I don't quite understand why every single piece (in particular qualitative stuff) needs to be explained. There is a vast amount of literature where you can get the information/proof yourself. And this is sth. which is the minimum what can be expected from a well rounded candidate.
Component VaR (CVaR): it is simply the marginal VaR times the wealth (W) invested in a particular asset and shows how the portfolio VaR would change if the component was deleted from the portfolio.
Porfolio volatility changes in a non-linear fashion due to different underlying components which drive the portfolio volatility (VaR). Component VaR is only useful when we have 1.) a large portfolio (high level of wealth invested) and 2.) having small individual positions (components). This ensures that the portfolio VaR changes more or less (approx.) proprionately if a component would be deleted from the portfolio.
CVaR can be sort of interpreted as risk measure to reflect
correlations because part of the CVaR calculation (formula) depends on the correl coefficient. Having a negative correl and therefore a negative beta, the position acts a hedge against higher portfolio VaR (in other words, it has a positive effect yielding to lower portfolio VaR).
Remember:
CVaR(asset A) = alpha * sigma(p) * W(A) * beta(A) =
CVaR(A)/VaR(p) =
w(a) * beta(A)
where the two on the right-hand side, yield the %-contribution of Asset(A) to total Portfolio VaR
1. CVaR(A)/VaR(p)
2. w(a) * beta(A)
- alpha denotes the z-value (will either be 1,65 at 95% or 2,33 at 99%)
- sigma (p) stands for portfolio volatility
- W(A) stands for total $wealth
- w(a) stands for %-weight of Asset(A)
To sum up, you can express the
component VaR either in dollar terms:
1. CVaR(asset A) = alpha * sigma(p) * W(A) * beta(A)
OR in %-terms of the portfolio VaR
1. CVaR(asset A) = CVaR(A)/VaR(p)
2. CVaR(asset A) = w(a) * beta(A)
Incremental VaR: change in the portfolio VaR due to adding a new position. In contrast to the component VaR it is also applicable for
large positions.
@David Harper CFA FRM do you agree with this?
In case of a large position, the portfolio VaR changes in non-linear fashion and IVaR is the better measure compared to the CVaR.
Put differently, IVaR is deemed to be more accurate when dealing with large (changes in) positions than the component VaR but comes with the drawback of doing a full revaluation (computionally burdensome).
We can achieve the approx. same result using either Marginal VaR or incremental VaR (
but only if the change in the position is small. Let's say you are invested in 2Million JPY and increase the position by JPY 10,000).
Notice:
- the marginal VaR is always very small in number (it only measures the tiny little change) whereas the component VaR is always big relative to the marginal VaR (the individual components of the portfolio should add up to the portfolio VaR approximately!) as the component VaR is the marginal VaR times the total wealth invested in asset (A)
- the component VaR can be written in %-form where we first need to determine the beta of the asset and then can write: component VaR = beta (asset A) * weight (in% of asset A)
Kevin Dowd' explanation:
Component VaR has an important limitation: it is a linear marginal analysis. The component risks add up to total VaR because of linear homogeneity working through Euler’s theorem, but the price we pay for this additivity property is that we have to assume that each component VaR is simply the position size multiplied by the marginal VaR. This is restrictive because it implies
that the component VaR is proportional to the position size: if we change the size of the position by k%, then the component VaR will also change by k%. Strictly speaking, this linear
proportionality is only guaranteed if each position is very small relative to the total portfolio; and where the position size is significant relative to the total portfolio, then the component VaR estimated in this way is likely, at best, to give only an approximate idea of the impact of the position on the portfolio VaR. If we want a ‘true’ estimate, we would have to resort
to the IVaR, and take the difference between the VaRs of the portfolio with and without the position concerned. The IVaR then gives us an exact estimate of the impact of the portfolio.
Unfortunately, this exactness has its price: we lose the additivity property, and the component VaRs no longer add up to equal the total VaR, which makes it difficult to interpret these IVaRs
(or CVaRs or whatever else we call them) as true decompositions of the total risk. In short, when positions are significant in size relative to the total portfolio, we can only hope for our
CVaRs to give approximate estimates of the effects of the positions concerned on the portfolio VaR.
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