3.4.1 General Ideas
A very useful tool in any kind of analysis is the use of incremental and marginal measures. An incremental measure is a comparison between two situations that can be arbitrarily close or distant. For example, if you have 50 shares of asset A and 100 of asset B and you come to have 90 shares of asset A and 180 shares of asset B, the question might be how to change the VaR, in other words tofind the incremental VaR. A marginal measure is an incremental measure where the increase becomes infinitesimal and in mathematics it is a derivative. Continuing with the previous example, with 50 shares of asset A and 100 of Asset B, the marginal VaR relative to asset A would be the difference between the current VaR and the VaR
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Estimated Probability
Portfolio Value at t = 1 (€)
Fig. 3.11 Probability distribution of the portfolio value attẳ1 (Author’s own composition)
corresponding to the portfolio whereby an additional infinitely small amount of money has been invested in shares of asset A.
Interestingly, an initial and afinal position must always be specified for an incremental measure, while in the case of marginal measures only the initial situation must be specified. This has consequences since, in prin- ciple, incremental measures compare two situations and it should be clear that these are the two options to choose from, while marginal measures indicate which steps must be taken to achieve the desired results. More precisely, it can be said that marginal measures are useful because their denotation, and to a greater extent their magnitude, directly indicate which procedure should be followed to achieve certain results. This idea will be clarified by the following example:
Suppose that F represents the VaR of a portfolio with four assets A, B, C and D and therefore depends on the amounts invested in each. Now imagine that the four marginal VaRs are calculated and0.5, 1,4 and 3 are obtained respectively. Taking into account that the higher the VaR is, the lower the risk becomes, from these measures the following conclu- sions, among others, can be reached: increasing the position of assets A and C increases the risk, while doing so in assets B and D reduces risk.
Furthermore, the reduction in risk arising from the position of D is three times that arising from asset B, meaning that for every euro invested in D to reduce the risk, three euros would need to be invested in B to obtain the same result. Similarly, the effect of lowering the risk due to asset B is twice that due to the increase in asset A, and therefore the position of A can be increased by a euro without increasing the risk, iffifty cents are invested in B.
Despite the previous explanations, it is very important to note the limitations of marginal measures due to the fact that these measures are local. In other words, they express what happens near a given point, that is, these measures, on being derivatives, are accurate when faced with infinitesimal changes, but as the changes become larger the results become increasingly inaccurate.6 This phenomenon is more important when dealing with functions which are less linear, as in the linear case it is
6This phenomenon has the same basis as the convexity phenomenon in the case of credit risk.
non-existent, and it should be kept in mind at all times that the VaR is strongly non-linear. If you have a portfolio with 50 currency units (cu) in shares of asset A and 100 cu in shares of asset B, the marginal VaR of this portfolio can shed a great deal of light on what will happen if 50.1 cu are invested in asset A and 100.4 cu in asset B, but not on what will happen with a portfolio of 100 cu invested in each asset, for which the incremental VaR is required. This is because the distance between the points (50,100) and (100,100) is very large, even though the latter components are equal.
Such problems do not occur with incremental measures, but as stated previously, they provide no insight on how to modify the starting point because they only compare situations.
3.4.2 Formal Definition
This section will proceed to establish the formal definition of incremental and marginal measures. For each risk measurement, and in fact for any type of measurement, its marginal measure can be defined as the partial derivative with respect to the corresponding amount. Thus, the marginal VaR can be defined as the derivative of the VaR with respect to a specific asset, and its economic interpretation is, as previously stated, the variation of the VaR of a portfolio on increasing the amount invested in an asset, which may or may not be part of the initial portfolio, by an infinitesimal amount of money.
As explained in the definition, if you have a portfolio made up of assets A and B with a given VaR, it is possible to calculate its marginal VaR with respect to asset A and also with respect to asset B, but the marginal VaR with respect to any other asset (C, D, etc.) can also be calculated simply by applying the previous definition. It should also be noted that the marginal VaR depends not only on the asset in which an additional amount will be invested, but also on the original portfolio—in fact, it depends on the correlation between both the original portfolio and the new asset and can be negative if the correlation between them is also negative.
Finally, it must be mentioned here that while the VaR is in the same currency unit as that of the portfolio value in question, the marginal VaR has no units because it is the derivative of the VaR with respect to the price
of a particular asset that can become a part of the portfolio and, therefore, has the same currency units.
Regarding the incremental VaR, it is defined as the difference between the VaR of two portfolios, the initial andfinal portfolios. This definition includes another similar measure, the component VaR, which is defined as the incremental VaR when the starting position does not have an asset and thefinal position does. More formally, if we consider a portfolio of N assets where d1,. . ., dN currency units are invested in assets 1,. . ., Nrespectively, the VaR component of asset Nis the VaR of a portfolio of Nassets where d1,. . ., dN currency units are invested in assets 1,. . ., Nrespectively, apart from the VaR of a portfolio ofN1 assets, all assets of the previous portfolio but the last (asset N), where d1,. . ., dN1
currency units are invested in assets 1,. . .,N1 respectively.
The usefulness of this measure lies in the fact that if a company hasN 1 investment projects to whichd1, . . ., dN 1 monetary units are dedi- cated respectively and is able to invest in a new project, to which they intend to devote dNcurrency units, the best way to ascertain the value of the investment and the benefits that the project could bring, in terms of risk, is to calculate the component VaR. The reason for this is that so far, the company's VaR is VaR (d1,. . .,dN1, 0) while the new VaR would be VaR(d1,. . .,dN), so the difference between both is the component VaR.
Thus, the component VaR measures the VaR variation owing to the new project and is a good tool to decide whether or not to go ahead with an investment; hence it is sometimes said that the component VaR is the contribution made by each component, or the element which changes the VaR on eliminating an asset.
Generalising this concept, these new incremental and marginal mea- sures are very useful to understand why different companies take on different projects. While the expected return and VaR of an investment are independent of the company that made the investment, the marginal VaR and incremental VaR, and in this case the component VaR, depend on the portfolio of existing assets before investing and, thus, such an investment involves different risks for different companies.
Finally, it must be mentioned that the incremental VaR, and therefore that of the component, should then be determined simply by calculating the VaR of the initial portfolio and the final portfolio. However, the
calculation of marginal measures is slightly more complicated, as it is based on the concept of a derivative. The easiest way to calculate this is to take two very close values and approximate the derivative using the formula f0ð ị x f xỵhð hịf xð ị, or a formula which is closer to reality, f0ð ị x f xỵhð ịf xh2hð ị.