TAKING VAR TO PIECES

C urrent methods of calculating value-at-risk prescribe either of two basic calculations: (a) the total, diversified VAR for a portfolio; or (b) the undiversified VAR for some subset of a portfolio. The portfolio subset might comprise all trades of a certain type or involving a certain asset, the individual trades themselves or even the solitary (mapped) cashflows. Except in rare and exceptional cases, however, the undiversified VAR numbers of the components of a portfolio almost never add up to the diversified VAR of the portfolio. Neither does the undiversified VAR provide any hint as to whether the corresponding components act to “hedge” the remainder of the portfolio or serve only to increase its risk. This leads us to the search for a useful definition of “component VAR” (CVAR). A good definition would have at least three properties: 0 if the components partition the portfolio (ie, are disjoint and exhaustive), then the CVARs should add up to the (diversified) portfolio VAR; C if the component were to be deleted from the portfolio, the CVAR should tell us, at least approximately, how the portfolio VAR will change; and, therefore Kl CVAR will be negative for components that act to hedge the rest of the portfolio. In this paper, we show that such a definition of CVAR may be based upon the VARdelta (or DelVAR) concept (Garman, 1996). VARdelta is a portfolio metric appropriate to the analytic (“variance-covariance”) methodology of VAR. The relationship of the VARdelta to the VAR is analogous to the relationship between the option delta and the option price. In this case, however, it measures the sensitivity of VAR to the injection of a unit of cashflow in each dimension (or “vertex”, as per JP Morgan’s RiskMetrics) of the cashflow space. Garman (1996) also shows how to analyse a new, candidate trade’s effect on portfolio VAR. Perhaps surprisingly, the same technique can be applied to trades already present in a portfolio, to form a useful and meaningful definition of CVAR.