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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?

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  • $\begingroup$ You more or less wrote down the definition of the Chi-squared distribution ... a weird example of a stock-market .. right? But thinking about variance (squared returns) it makes sense.. $\endgroup$ – Ric Oct 13 '14 at 20:16
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Its Chi-Square distribution ($k=$ number of portfolio assets): http://en.wikipedia.org/wiki/Chi-squared_distribution#Definition

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  • $\begingroup$ What do you mean by setting $k=1$? $\endgroup$ – Hans Oct 12 '14 at 0:28
  • $\begingroup$ @Hansen Its when you have a variable $k$, it can have a value $1$. $\endgroup$ – emcor Oct 12 '14 at 0:30
  • $\begingroup$ $k$ is the number of stocks in the portfolio and can take on any natural number. It is just a bit strange that you set $k=1$. Are you trying to give an example for when $k=1$ as opposed to, say, $k=217$? $\endgroup$ – Hans Oct 12 '14 at 0:53
  • $\begingroup$ @Hansen Yes sry, I misread from the question $X_1^2$, but I suppose its $X_i^2$. $\endgroup$ – emcor Oct 12 '14 at 1:10
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Yes. It is a simple transformation of the product of the Gaussians from the Cartesian to the hyper-spherical coordinate.

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