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

*-th asset in the portfolio, and*

**i***, or*

**V***, is total portfolio value in dollars. Now assume*

**V**_{p}*is the return on asset*

**r**_{i}*for period t, and is represented as a column vector of returns.*

**i**Portfolio returns can now be calculated as * w^{T}r*, where

*is the transpose of*

**w**^{T}*. In other words:*

**w**We also know that if * V* is the covariance matrix for the portfolio, then . Where

*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.*

**σ**_{p}# Calculating portfolio VaR

Since we know the portfolio standard deviation, VaR is easy to calculate as a product of the standard deviation and the z-score from the normal distribution for the confidence level we seek. (eg, at 95% confidence, or if * α* = 5%, then =NORMSINV(0.95) in Excel gives us the value for

*).*

**z**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

The VaR calculation above takes into account the correlations between the assets in the portfolio. Undiversified VaR is VaR calculated as a summation of the VaRs of each individual asset. Undiversified VaR is therefore generally much larger than regular diversified 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

*-th asset, then*

**i***can be calculated as the derivative of VaR with respect to*

**MVaR**_{i}*. Or:*

**V**_{i}We know what * VaR_{p}* is (

*), and*

**=zσ**_{p}V*can be written as*

**V**_{i}*, so substituting we get:*

**V**_{p}w_{i}* —–Equation 1*

Now we know that

or

Differentiating this wrt * w*, we get

The * i*-th 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

We can also calculate the beta of the individual assets in the portfolio, which is the sensitivity of the returns between the i-th asset to portfolio returns.

# Component VaR (CVaR)

Component VaR for the i-th asset is nothing but the product of Marginal VaR and the value of the i-th asset. Component VaR has the useful property that it adds up to the dollar VaR of the portfolio, that makes life very easy from a risk disaggregation perspective.

And remember that CVaR totals to VaR.