If I would like to construct a fully invested long-only portfolio with two asset classes (Bonds $B$ and Stocks $S$) based on the concept of risk-parity.
The weights $W$ of my portfolio would then be the following:
Then the weight of the bonds: $$W_B = \textrm{Vol}(S)/[\textrm{Vol(S)}+\textrm{Vol(B)}]$$
and the weights of the stocks $$W_S = 1 - W_B$$
Based on this result, I am going to overweight the low-volatility asset and underweight the high-volatility asset.
My question is: how do I calculate the weights for a portfolio with multiple asset classes, 5 for example, so that each asset class will have the same volatility and contribute the same amount of risk into my portfolio. From historical data I can extract the volatility of each asset class and the correlation between them.