Aggregating VaR on a Multilevel Tree

[IAmEric]

Hello,



I've got a multilevel tree with a single root node at the top. This root node has N children nodes representing various risk factors. The NxN correlation matrix among these risk factors is known. Each child node (i) may itself be a parent node having N(i) children representing "sub" risk factors. The correlation matrix for the children of each independent parent is known.



To say it differently, I have a multilevel tree and for each parent node I have a distinct correlation matrix representing the correlations of the siblings.



At each lowest level "leaf" node, we know the VaR. If we assume normal distributions, the VaR is proportional to the volatility and we can aggregate the VaR to the parent VaR with our knowledge of the correlations.



VaR_parent = sqrt(sum_ij VaR_child(i) VaR_child(j) rho_ij)



Since the correlations are known for each parent node, we can aggregate the risk all the way up to the root node using the above expression iteratively.



My goal is to take this general idea and extend it to the case where the risk factors may not be normally distributed using Monte Carlo. I'm finding it surprisingly difficult to do this in an elegant way. As a first run, I simply want to reproduce the results of the above method, but using MC instead of matrix multiplications.



Is this a lost cause? Is there a simple way to do this?



To be repetitive in the hopes of clarity, the problem is essentially to determine the volatility at the root node of a multilevel tree where each parent node has its own correlation matrix and the volatilities of the leaf nodes are known.



Thanks for any words of wisdom Beer

Eric

[Nonius]

hmmm, when we MCed VaR for the swiss bank, we had to aggregate VaR up a risk hierarchy as well....no, I don't think you will have nice formulae for the aggregation in the general setting of nonlinear portfolio OR non-normal risk factors, and in fact this aggregation stuff is what slows everything down so much.....

[sfca]

I'm not sure what you are doing but I get confused easily.  You want a different risk factor at the security level and are trying to aggegate all the way up?  If thats the case, you might want to do the reverse and have a (far fewer) number of primative risk factors at the aggregate level and map each security to those factors.  I believe thats the RiskMetrics approach.

[Nonius]

well, I think that the general construction is you have a hierarchy that describes the granularity in which you want to view value at risk.  inductively, you have a node that represents something (a risk factor, a business unit, a country branch, etc), and you want to know VaR at that node.  each node, except for the lowest nodes, have children.  There is an operation, called "Aggregate", assigns to the father node a VaR number as a function of everything that was used to caluculate VaR for the children + some assumption of dependency.  In the case of normal assumption plus linear portfolios, Aggregate boils down to a matrix calculation.  In general, however, it is much more complicated because you can't just use the VaR numbers at the children's nodes as input into Aggregate.  In other words, The relation between children and father via Aggregate, is implicit, rather than explicit.

[IAmEric]

I am dead sick, but was obsessing about this all night in a groggy haze Dead



I think that its actually not that difficult to aggregate the risk up the tree if you keep everything in terms of P&L (just simulate N runs and grab the desired quantile at each level), but there is one fairly major modification that may be needed to do it via MC. As the problem was stated originally the correlation matrix at each parent was the correlation of the absolute P&L of each sibling. I am thinking about modifiying this so that the correlation at each parent node is the correlation of the [i]spreads[/i] of the children with respect to some baseline of the parent.



This way (which may be what sfca was suggesting) you can work your way down the tree, i.e.



PL_child = BasePLofParent + Spread_child



You simulate the baseline P&L of the parent and simulate spreads for the children separately and simply add the spreads to the baseline to get the child P&L.



How does that sound?



Thanks,

Eric

[Nonius]

if you are doing non-linear portfolios, chuck the correlations....you just have to add the PLs at each father node for each simulation ID, and then re-do the quantile calculation.  that turns out to be a slow process, or at least, I found that when your tree as big as the Alps.

[IAmEric]

Excellent. Thank you.



For my purposes, I think what I outlined will be good enough for multivariate normal risk factors. The nonlinearity will come in through the sensitivities to those risk factors, i.e. at the leaf nodes, I will have a bunch of correlated risk factors, these get mapped through a nonlinear sensitivity to a P&L at each leaf node. Then we just add up the P&Ls as we move up the tree. Yeah, I can see how this would be slow if you had a huge tree, but my tree is not that big so hopefully this will be alright. If I start out with linear sensitivities, then all the P&Ls will also be normal and I can check the result via matrix manipulation.



Thanks,

Eric