# Marginal, Incremental and Component Value at Risk (VAR)

In the previous articles, we have studied how the value at risk (VaR) models should be used to calculate risks. However, the calculation of risks is not the ultimate end goal. Instead, it is efficient management of this risk that makes the value at risk (VaR) model valuable.

Let’s assume that in a particular instance, the value at risk (VaR) model gives a value that is unacceptable to the organization. In such a situation, the organization would ideally have to divest a part of its portfolio in order to reduce the overall risk. Here it will be faced with the question about which part of their portfolio should be divested. Hence, the risk characteristics of sub-portfolios present within the portfolio will have to be measured in order to decide which part should be divested.

Such calculations are done using marginal, incremental, and component value at risk (VaR) models. These models allow the overall risk to be broken into its component parts for the purpose of analysis.

In other words, these models are used to drill down the overall value at risk (VaR) number into its constituent parts. In this article, we will have a closer look at what these statistics are and how they should be interpreted.

Define sub-portfolio: The first step in conducting and kind of drill-down analysis is to clearly define component parts. These component parts are then labeled as sub-portfolios and all the statistical analysis is done at a sub-portfolio level. Different organizations define their sub-portfolios in different ways.

Some organizations define their sub-portfolios at an asset class level. This means that the stocks will be considered to be one sub-portfolio, whereas bonds will be considered another one and derivatives will be considered a third one.

Similarly, it is also possible to define a sub-portfolio at the line of business level. For instance, corporate banking can be considered to be one sub-portfolio, investment banking can be considered to be a second portfolio whereas retail banking can be considered to be the third one.

Once the component parts have been decided, the next step is to break the overall value at risk (VaR) into these components.

1. Marginal Value at Risk (VaR): The purpose of marginal value at risk (VaR) is to find out the risk each sub-portfolio is adding to the overall portfolio. Hence, the first step begins by assigning a dollar value to each sub-portfolio.

The next step is to completely eliminate one sub-portfolio and then recalculate the value at risk (VaR).

For instance, if a portfolio consists of stocks, bonds, and derivatives, we can remove derivatives and then calculate the value at risk (VaR) for stocks and bonds. The difference between the two value-at-risk (VaR) numbers will be the VaR for derivatives. This analysis is very useful since it explains how much risk each sub-portfolio is adding to the overall risk. This analysis then becomes the basis for hedging and risk management efforts. The divestment of portfolios is also often done on this basis.

2. Component Value at Risk (VaR): The concept of component value at risk (VaR) is linked to the concept of marginal value at risk (VaR). In fact, component value at risk (VaR) can be thought of as being value at risk (VaR) expressed in a dollar amount. The component value at risk (VaR) is calculated by finding the weight of the position being deleted from the overall portfolio.

For instance, if assets worth \$250 were being deleted from a \$1000 portfolio, then the weight assigned would be 25%. This weight is then multiplied by the marginal value at risk (VaR) and the portfolio value of that position to be deleted.

3. Incremental Value at Risk: The incremental value at risk (VaR) method is often confused with the marginal value at risk (VaR) method. However, they are different from one another.

In the case of incremental value at risk (VaR), none of the sub-portfolios are completely eliminated. Instead, small changes are made to the values of these portfolios and the resultant value at risk (VaR) is calculated. There are two methods that are commonly used to calculate the incremental value at risk (VaR).

1. The first method is called the full valuation approach. Simply put, this means that the value of the entire portfolio is calculated once again. For instance, it is assumed that the value of the entire portfolio has been increased by 1%. As such, the entire value at risk (VaR) is calculated once again. The newly calculated value at risk (VaR) is then subtracted from the original value at risk (VaR). The residual value is the value of a 1% increment in the portfolio.

2. The full valuation method may seem easy to use. However, there are several issues with using this approach. Firstly, the portfolios of large organizations typically consist of hundreds of assets. Hence, calculating the entire value at risk (VaR) again can be tedious and time-consuming. Hence, another method called the approximate solution approach is used. The approximate solution approach is a shortcut method that uses statistical methods to calculate value at risk (VaR) without losing the efficiency.

The purpose of incremental value at risk (VaR) is to explain how much additional risk is added if we increase another unit of the portfolio.

The bottom line is that marginal, component, as well as incremental value at risk (VaR), provide market risk managers with an important tool for understanding the root cause of the market risk. This helps risk managers identify the sub-portfolios which are creating disproportionately high risks and then eliminate them.

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