I am trying to understand an optimization algorithm to achieve risk parity in a portfolio. I need some help figuring out the notation in the following formula:

Formula, algorithm

I found this on THIS paper.

I understand the following, if you could help me by pointing any mistake, would be great!

I understand that this algorithm is suppossed to iterate the allocation for each asset at a time.

  • $x^*_i$ : The iteration n+1 of asset i.

  • $σ_i$ : The standard deviation of Asset i

  • $x_j$ : allocation for each asset j

  • $\sigma_j$ : The standard deviation of asset j

  • $\rho_{i,j}$ : This is my biggest question. WHAT is this?

  • $b_i$ : The risk budget for the asset, which for risk parity is $\frac{1}{n}$

  • $\sigma(x)$ : The standard deviation of the portfolio

What am I missing?

  • $\begingroup$ @AlexC I know the answer is very simple, but please don't answer in comments we will never get these type of questions closed otherwise... $\endgroup$
    – SRKX
    Commented Jun 16, 2016 at 1:21
  • $\begingroup$ The link to the paper is broken. I suspect that the capture is from arxiv.org/abs/1902.05710 ? $\endgroup$
    – jChoi
    Commented Nov 24, 2021 at 4:44

1 Answer 1


Your question seems very simple. The $\rho_{ij}$ are the correlations between asset i and asset j, in other words these are the elements of the correlation matrix. This notation is very standard in portfolio optimization problems. The number of securities n, the n-by-n correlation matrix R and the n vector of $\sigma_j$'s are the main inputs of a risk parity problem.

  • $\begingroup$ Great! Thank you very much. I was able to solve it after all! $\endgroup$ Commented Jun 18, 2016 at 20:51

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