I am given the following data:
The question asks to determine the investment that is driving the VaR, and I am not quite sure what it means to "drive" the VaR. Any help is appreciated!
If you have a covariance matrix, $Q$ the VaR is a measure of the standard deviation of the portfolio,
ie. $$VaR, V \propto \sqrt{S^T Q S}$$
and,
$$ \frac{\partial V}{\partial S} = \frac{QS}{V} $$
Suppose you had 3 assets, with large positions in the first two assets, and small position in the third, AND that the first two were perfectly negatively correlated, ie.
$$ Q = \begin{bmatrix} 1 & -1 & 0\\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$
$$ S = \begin{bmatrix} 100 \\ 100 \\ 1\end{bmatrix} $$
Then your VaR from the formula above is 1, and intuitively you can allocate it completely to the third instrument.
If you use an inclusion exclusion method your VaR will be completely different.
However, the derivative:
$$ \frac{\partial V}{\partial S} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} $$
which gives the intuitive proportional expected allocation. I believe this is the equivalent to an Aumann-Shapley allocation principle.
** Edit **
This approach does not necessarily help with the OP question where only specific information is available, i.e. historical PnL vector of portfolio, which of course contains no information at all with which to allocate VaR to sub positions, and then the specific PnLs of all individual instruments on the worst 10 days. I suspect you may be able to hypothesis some form of allocation procedure from this information but given it is a sample of only 10 days worth of data it will be subject to large variance (uncertainty) and therefore not necessarily useful/reliable.
It sounds like the P&L's you are given are not really the historical P&L's.
Rather, you have some portfolio and market data currently; you have 260 days of historical market data changes; and you calculate what the P&L of the present portfolio would have been if the market moved as it did on that historical date from the current market data.
You're not given the helpful information that in real life would have been useful in your investigation : the market risks for the portfolio (ideally for each position) and P&L explain's for each historical scenario. It sounds like you don't even know what the historical market scenarios are. In real life, most of this data would be readily available to you.
If the VaR is 99%, then it corresponds to the 260*(1-99%)= 2nd or 3rd worth P&L under the historical scenarios.
You're given some additional details about the 10 days with the worst P&L - the P&L of each investment.
If each "investment" includes its market hedges, then you simply identify which of the investments caused most of the negative P&L on the historical date that was chosed for the VaR. That's what drove the VaR!
Additionally (not quite part of the question posed to you) it would also be prudent to see whether this historical scenario was a one-day fluke, or approximately the same investments had bad P&L in the other 9 worst scenarios.
If the "investments" are individual positions that are part of larger strategies, then this analysis is not very useful. Maybe some position lost money but its hedge did its job?
If you can group the individual positions with their hedges and split the portfolio into logical strategies and their hedges, stuff like "EUR gamma scalping" or "Ukraine credit" (if some position hedges multiple strategies, split the hedge between the strategies.) and do the above - figuring which strategies had bad P&L under the VaR scenario.
One way to look at answering this question is VaR Contribution.
Evaluate VaR of the Portfolio, and then evaluate VaR of the Portfolio without the asset. The largest difference of VaR with the asset - VaR of the portfolio without the asset would be the asset which is contributing the most to VaR.
You may want to correct the size of the portfolio for each assets exclusion if you are calculating dollar VaR.
Here are a couple of links that describes marginal contribution and component VaR that might help you understand:
https://www.investopedia.com/terms/m/marginal-var.asp https://www.bionicturtle.com/forum/threads/individual-var-vs-component-var.1373/
edit
In taking a closer read of your question and the data you have available, you would not be able to do any of these approaches. You would neither be able to generate a variance-covariance matrix nor brute force it with exclusion/inclusion. It sounds like what they are asking for is the largest contributor to the historical VaR and can be answered doing the following:
take the (1-var confidence interval)/2, (95% CI var is very common) = (1-.95)/2 = 2.5%. This is the percent of historical losses that will be in the tail of worst losses for the 95% var.
multiply 2.5% * 260 observations you have been given = 6.5. Therefore take the 7th day of worst returns and find the asset that had the worst PnL for that observation.
The portfolio or correlation effect will be accounted for in that you are using actual historical portfolio returns. Also, as you are using actual historical returns, you are not making any assumptions about any parameters or distributions of returns.
VaR is a loss function calculated from what's available in step 1, whose value is a magnitude, and whose sign indicates whether there is a portfolio loss, or a negative loss (which is actually a gain, given that VaR, as a loss, is ordinarily reported as a negative number).
So to ask which asset, whose returns are available in step 2, is driving this loss function is just the same as asking "which asset is contributing most to the magnitude of VaR" (more likely in the loss direction). in other words, which of the assets is the largest risk contributor.
Although the question doesn't ask for a calculation of risk (VaR) contribution, I can say that it can be done without inclusion or exclusion of assets one-at-a-time as what the other answer suggests, but rather through partial derivatives of total VaR with respect to individual assets if the covariance approach is used, since this would retain the dependence/correlation structure of the portfolio
