Converting Contribution To Risk from Variance to Stdev

Question

So we have the basic structure:

$\sigma^2_{Pxy} = w_x^2 \sigma_x^2 + w_y^2 \sigma_y^2 + 2 w_x w_y \sigma_{xy}$

$\sigma^2_{Px} = w_x^2 \sigma_x^2 + w_x w_y \sigma_{xy}$

$\sigma^2_{Py} = w_y^2 \sigma_y^2 + w_x w_y \sigma_{xy}$

$\sigma^2_{Pxy} = \sigma^2_{Px} + \sigma^2_{Py}$

The problem is that this structure leaves the risk measure in terms of variance, so if we want to find percentage contribution to risk, it doesn't easily translate from variance to standard deviation.

Meaning, it is easy to solve for:

$ \cfrac{\sigma^2_{Px}}{\sigma^2_{Pxy}} =$ percentage contribution to risk (variance), as

$ \cfrac{\sigma^2_{Px}}{\sigma^2_{Pxy}} + \cfrac{\sigma^2_{Py}}{\sigma^2_{Pxy}} = 1$

But, it is not easy to solve for:

$ \cfrac{\sigma_{Px}}{\sigma_{Pxy}} =$ percentage contribution to risk (std dev), as

$ \cfrac{\sigma_{Px}}{\sigma_{Pxy}} + \cfrac{\sigma_{Py}}{\sigma_{Pxy}} \ne 1$

Additionally, it isn't right to just scale it to whatever the square root of the percentages happen to sum to.

If $ x = \cfrac{\sigma_{Px}}{\sigma_{Pxy}}$ and $ y = \cfrac{\sigma_{Py}}{\sigma_{Pxy}}$, it is not clear what $ \cfrac{x}{x + y}$ equals.

Does anyone have a solution to this problem?

Answer 1

$ \cfrac{\sigma_{Px}}{\sigma_{Pxy}} + \cfrac{\sigma_{Py}}{\sigma_{Pxy}} \ne 1$

beacuse

$ {\sigma_{Px}} + {\sigma_{Py}} \ne {\sigma_{Pxy}}$

as we had

$\sigma^2_{Px} + \sigma^2_{Py} = \sigma^2_{Pxy}$

and this is equal to

$\sqrt{(\sigma^2_{Px} + \sigma^2_{Py})}/\sigma^2_{Pxy} = 1$

Let me know if you want to see an example of risk contribution based on standard deviations of the constituent assets.

UPDATE:

Since

$\sigma^2_{Pxy} = w_x^2 \sigma_x^2 + w_y^2 \sigma_y^2 + 2 w_x w_y \sigma_{x} \sigma_{y} \rho_{xy}$

the standard deviation of the portfolio is

$\sigma_{Pxy} = \sqrt{w_x^2 \sigma_x^2 + w_y^2 \sigma_y^2 + 2 w_x w_y \sigma_{x} \sigma_{y} \rho_{xy}}$

Now find the marginal risk contribution of asset $x$ by taking the derivative of the portfolio standard deviation with respect to $w_x$ (the weight of asset $x$):

$\cfrac {d \sigma_{Pxy}} {d w_x} = \cfrac {d \sqrt{w_x^2 \sigma_x^2 + w_y^2 \sigma_y^2 + 2 w_x w_y \sigma_{x} \sigma_{y} \rho_{xy}}} {d w_x} = \cfrac {d (w_x^2 \sigma_x^2 + w_y^2 \sigma_y^2 + 2 w_x w_y \sigma_{x} \sigma_{y} \rho_{xy})^{1/2}} {d w_x} = \cfrac {1} {2 (w_x^2 \sigma_x^2 + w_y^2 \sigma_y^2 + 2 w_x w_y \sigma_{x} \sigma_{y} \rho_{xy})^{1/2}} (2 w_x \sigma_x^2 + 2 w_y \sigma_{x} \sigma_{y} \rho_{xy}) = \cfrac {w_x \sigma_x^2 + w_y \sigma_{x} \sigma_{y} \rho_{xy}} {\sigma_{Pxy}}$

Total risk contributed by asset $x$ to the portfolio is equal to

$\cfrac {w_x \sigma_x^2 + w_y \sigma_{x} \sigma_{y} \rho_{xy}} {\sigma_{Pxy}} w_x = \cfrac {w_x^2 \sigma_x^2 + w_x w_y \sigma_{x} \sigma_{y} \rho_{xy}} {\sigma_{Pxy}}$

Likewise, total risk contributed by asset $y$ to the portfolio is

$\cfrac {w_y \sigma_y^2 + w_x \sigma_{x} \sigma_{y} \rho_{xy}} {\sigma_{Pxy}} w_y = \cfrac {w_y^2 \sigma_y^2 + w_x w_y \sigma_{x} \sigma_{y} \rho_{xy}} {\sigma_{Pxy}}$

Example: $\sigma_{x} = 15%$; $\sigma_{y} = 20%$; $\rho_{xy} = 0.5$; $w_x = 60%$; $w_y = 40%$;

So we have: $\sigma_{Pxy} = 14.73%$;

Risk contributions to the portfolio: $\sigma_{Px} = 7.94%$; $\sigma_{Py} = 6.79%$;

Is this what you were looking for?