Allocating Portfolio Volatility to Individual Assets
Summary
The document considers how to attribute the volatility of an equally weighted portfolio of three stocks to its components, given their individual volatilities and pairwise correlations. It distinguishes incremental risk, measured by comparing portfolio volatility with and without an asset, from Euler contributions to volatility. Incremental contributions are nonlinear and do not generally add up to total portfolio volatility.
For an additive allocation, the response assigns each asset a contribution equal to its weight times its covariance with the portfolio return, divided by total portfolio volatility. Summing these terms recovers portfolio volatility; equivalently, each contribution is the asset weight multiplied by the marginal change in portfolio volatility as that weight varies. The document provides the general allocation formula but does not calculate numerical contributions for the example. The approach allocates volatility under the stated portfolio return and covariance assumptions; it is not the only possible notion of risk contribution.
Key ideas
- Incremental risk compares portfolio volatility with an asset against volatility after replacing that asset with cash.
- Incremental risk contributions are nonlinear and do not generally sum to total portfolio volatility.
- Euler allocation defines an asset’s volatility contribution using its weight and covariance with portfolio returns.
- The Euler contributions sum to portfolio volatility and equal each weight times its marginal volatility contribution.
- The document gives a formula but does not compute the example’s numerical allocations.
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# How to determine portion of portfolio's risks from components?
# How to determine portion of portfolio's risks from components?
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 by Richi Wa (score 2)
https://quant.stackexchange.com/a/25763
You can do 2 things:
- 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 $\sigma$.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.