10.2 Contribution to Risk
Volatilities and variances do not add and equation (10.1) does not, on the surface, provide a decomposition of portfolio volatility into contributions due to individual assets or groups of assets. Nonetheless, there are two useful ways we can define the contribution a position makes to the volatility or VaR:
1. Infinitesimal: change in volatility or VaR due to an infinitesimal change in a position.
2. All-or-nothing: change in volatility or VaR due to complete removal of a position.
In my view, the infinitesimal, or marginal contribution to risk, and the decomposition it provides, is one of the most powerful but underappreciated tools for risk analysis. Such a contribution to risk provides a useful decomposition of the current risk profile by showing how the current positions affect the current portfolio, aiding in the understanding of the portfolio. Positions in a portfolio are usually adjusted little by little rather than by complete removal of a position, and the marginal contribution provides a good estimate of this for a large portfolio with many small positions. I find the infinitesimal, rather than the all-or-nothing measure, to be far the more useful. Although the change due to complete removal of an asset (setting to zero position) is valuable information, I think the best hedges analysis, discussed further on, is generally more useful.
Unfortunately, there is no agreement in the literature, and considerable confusion, regarding nomenclature, ...