Risk Parity portfolio construction

Question

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 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 I am going to overweigh the low volatility asset and underweight the high volatility asset. My question is: How to 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.

Answer 1

Risk Parity is not about "having the same volatility", it is about having each asset contributing in the same way to the portfolio overall volatility.

The volatility of the portfolio is defined as:

$$\sigma(w)=\sqrt{w' \Sigma w}$$

The risk contribution of asset $i$ is computed as follows:

$$\sigma_i(w)= w_i \times \partial_{w_i} \sigma(w)$$

You can then show that:

$$\sigma(w)=\sum_{i=1}^n \sigma_i(w)$$

The vector of the marginal contributions ($\partial_{w_i} \sigma(w)$) is computed as follows:

$$c(w)= \frac{\Sigma w}{\sqrt{w' \Sigma w}}$$

You can then find the solution by running the following optimization:

$$\underset{w}{\arg \min} \sum_{i=1}^N [\frac{\sqrt{w^T \Sigma w}}{N} - c(w)_i]^2$$ This article contains all the developments you require to understand how the formulas above are derived.