How to determine portion of portfolio's risks from components?

Question

Say I have a portfolio of 3 stocks $A,B,C$ with $\mu_A = 5%$, $\mu_B = 10%$, $\mu_C = 15%$ and volatility $\sigma_A = 10%$, $\sigma_B = 15%$, and $\sigma_C = 25%$. Let us also say that correlations are $\rho_{AC} = 0.7$, $\rho_{AB} = 0.3$, and $\rho_{BC} = -0.1$. Say total portfolio value is 1 and it is composed of $A,B,C$ equally by value. How would I calculate the corresponding risk exposure that I have to each of the three underlying securities?

Portfolio $\mu_{total} = \frac{1}{3} \times \mu_A + \frac{1}{3} \times \mu_B + \frac{1}{3} \times \mu_C$.

Portfolio $\sigma_{total} = \sqrt{\frac{1}{9}(\sigma_A^2+\sigma_B^2+\sigma_C^2 + 2\rho_{AC}\sigma_A\sigma_C+2\rho_{AB}\sigma_A\sigma_B+2\rho_{BC}\sigma_B\sigma_C)}$

How would you divide up $\sigma_{total}$ or is it not possible?

Answer 1

You can do 2 things:

1. incremental risk: Calculate the volatility with the asset and with the asset replaced by cash. The difference gives you the (non-linear) incremental risk contribution of the asset. They don't sum up to $\sigma$.

2. contributions to volatility (Euler allocation)

As $\sigma = \sigma^2/\sigma$ you can define risk contributions by $$ \frac{w_i cov(r_i,r)}{\sigma}, $$ where $w_i$ is the weight of asset $i$, $r_i$ is its return (as a random variable) and $r$ is the portfolio return (with the asset $i$ weighted by $w_i$). If you add these quantities: $$ \sum_{i=1 }^n \frac{w_i cov(r_i,r)}{\sigma} = \frac{\sum_{i=1 }^n w_i cov(r_i,r)}{\sigma} = \frac{cov(\sum_{i=1 }^n w_i r_i,r)}{\sigma} = \sigma^2/\sigma = \sigma. $$ This can also be seen as $w_i \frac{\partial \sigma}{\partial w_i}$ where $\frac{\partial \sigma}{\partial w_i}$ is called the marginal contribution to volatility.