Links to all tutorial articles (same as those on the Exam pages)VaR disaggregation  Marginal and Component VaR
This article covers how Marginal and Component VaRs are calculated. We follow up (in a separate article) with a real life example of how VaR, MVaR, undiversified VaR, Component VaR are calculated  based on actual price data pulled from Quandl, an open source market data website.
Imagine i assets in the portfolio, each with a weight of w_{i}.
Just another way of saying that the sum of weights equals one. The weights can be represented as a column matrix, w, as below:
Also assume that V_{i} represents the dollar value of the ith asset in the portfolio, and V, or V_{p}, is total portfolio value in dollars. Now assume r_{i} is the return on asset i for period t, and is represented as a column vector of returns.
Portfolio returns can now be calculated as w^{T}r, where w^{T} is the transpose of w. In other words:
We also know that if V is the covariance matrix for the portfolio, then . Where σ_{p} is the portfolio variance. For more on how we get this, and how the matrix math works, read the article on portfolio variance also posted in the tutorials.
Calculating portfolio VaR
Note that this is VaR as a percentage. To calculate the dollar VaR, this will need to be multiplied by the value V of the portfolio.  in dollars.
Undiversified VaR
Marginal VaR for asset i
Marginal VaR for an asset i in the portfolio is the change in VaR caused when an additional $1 of the asset is added to the portfolio. Mathematically, if V_{i} is the value of the ith asset, then MVaR_{i} can be calculated as the derivative of VaR with respect to V_{i}. Or:
We know what VaR_{p} is (=zσ_{p}V), and V_{i} can be written as V_{p}w_{i}, so substituting we get: Equation 1
Now we know that or
Differentiating this wrt w, we get
The ith component of this is calculated as follows (more of the math is explained in Chapter 3, Vol III Book 3 of the PRMIA Handbook):
Substituting this in Equation 1 (in the last term), we get: Equation 2
Even if you do not understand the calculus, that is okay so long as you know the above result and can apply it to a numerical question.
MVaR can also be expressed in terms of correlation (as opposed to covariance), and beta by taking advantage of the following relationships:
Substituting these appropriately, we get: Equation 3 and Equation 4
Try to remember all the above equations numbered 1 to 4 above to prepare for any numerical questions relating to MVaR.
Calculating beta
Component VaR (CVaR)
And remember that CVaR totals to VaR.
