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Attributing Portfolio Value at Risk: Relations with Component VaR, Marginal VaR, and Incremental VaR

Consider a simple portfolio worth , consisting of two assets. Its value at risk, , depends on the initial position size of one of the assets (blue arrow). A hypothetical change in the position size of this asset will trace out a new hypothetical portfolio, plotted as the blue curve.
The difference in portfolio VaRs before and after the position change is known as the incremental VaR (red arrow): . The incremental VaR gives the amount of risk that a new position adds to a portfolio. It measures the impact of a new trade on the overall portfolio risk. It is related approximately linearly to its beta: .
The marginal VaR measures the sensitivity of portfolio VaR to a small change in an asset holding (slope of gray triangle): .
The component VaR (purple arrow) of an asset measures the contribution of this asset to portfolio VaR. It depends linearly on the marginal VaR: .
The component VaR attributes portfolio VaR to its constituent assets. In other words, the component VaRs are mutually exclusive and exhaustive partitions of portfolio VaR: . It is this additive property that makes risk allocation possible: the risk of the whole firm is equal to the sum of the risks of its constituents. In contrast, because of diversification benefit, incremental VaR does not enjoy this additive property.
The magnitude and sign of component VaR show where risk is concentrated: which positions constitute big risks and which positions can serve as natural hedges.
Because the component VaR is based on a linear approximation (), it is more accurate when the portfolio is well diversified, such that each position size is vanishingly small () relative to the portfolio size.
As a tool for risk budgeting, risk allocation involves a choice. On the one hand, the incremental VaR is accurate but nonadditive. On the other hand, the component VaR is less accurate for an undiversified portfolio, but its additivity conserves portfolio VaR.

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DETAILS

Snapshot 1: , as a measure of VaR's sensitivity to small changes in portfolio holdings, is less reliable where VaR is nonlinear, seen as a large divergence between the blue portfolio VaR curve and its linear interpolation (slope of gray triangle)
Snapshot 2: portfolio has no diversification benefit (vanishing ) when assets are perfectly correlated (), in which case attains its maximum value
Snapshot 3: diversification benefit is maximized (largest ) when assets are perfectly contrarian (), in which case attains a minimum value
Reference
[1] P. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed., New York: McGraw-Hill, 2001 pp. 153–163.
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