# 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 $1 piece from each stock? Will it be simple the Chi-squared distribution with parameter k? • You more or less wrote down the definition of the Chi-squared distribution ... a weird example of a stock-market .. right? But thinking about variance (squared returns) it makes sense.. – Ric Oct 13 '14 at 20:16 ## 2 Answers Its Chi-Square distribution ($k=$number of portfolio assets): http://en.wikipedia.org/wiki/Chi-squared_distribution#Definition • What do you mean by setting$k=1$? – Hans Oct 12 '14 at 0:28 • @Hansen Its when you have a variable$k$, it can have a value$1$. Oct 12 '14 at 0:30 •$k$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,$k=217$? – Hans Oct 12 '14 at 0:53 • @Hansen Yes sry, I misread from the question$X_1^2$, but I suppose its$X_i^2\$. Oct 12 '14 at 1:10

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