Incremental Value at Risk
Incremental value at risk, or iVaR, is a measure of risk attribution. It tells us how much risk a position or sub-portfolio is adding to a portfolio. It can be positive or negative. If the iVaR of a position is positive then increasing the size of the position slightly will increase the value at risk of the portfolio. Likewise, if the iVaR is negative then increasing the size of the position slightly will lower the value at risk of the portfolio.
Mathematically, if the ith position in a portfolio has exposure or weight, wi, then the iVaR of that position relative to the portfolio is:
It may be easier to get an intuition for iVaR by rearranging this equation as:
The first term on the right-hand side, dwi/wi, can be thought of as the percentage change in the position size. If we have $200 of a security, and we add $2 to the position, then dwi/wi is 1% = $2 / $200. The left-hand side of the equation, d(VaR) is just the change in the VaR of the portfolio. In other words, if we change the size of a position in our portfolio by 3%, then the VaR of the portfolio will change by the iVaR of that position multiplied by 3%. Equation 2 is really only valid for infinitely small changes in wi, but for small changes it can be used as an approximation. Using iVaR in this way is similar to using delta to approximate the change in value of an option.
Incremental value at risk is additive. The sum of the iVaR of all of the positions in a sub-portfolio will equal the iVaR of that sub-portfolio. What’s more, the sum of the iVaR of all of the positions in a portfolio will equal the VaR of that portfolio (in effect, the iVaR of a portfolio to itself is its VaR).