Estimating component and marginal TVaR…(s) is a computationally intensive task in the capital allocation process. These two risk metrics are actually convergent, please see a four equations proof on the blog picture.
We call standalone TVaR, the risk metric computed for each risk independently from the rest of the portfolio. We define component TVaR as the metric back-allocated from the total portfolio TVaR. With back-allocation it contains covariance risk, as it should, since the Portfolio TVaR takes into account inter-risk dependencies. In Equation (1) we back-allocate the total portfolio metric using a stand-alone TVaR ratio to the portfolio risk components, - i.e. this gives us our component TVaR…(s). These are cumulative and sub-additive as Equation (2) shows. Then in Equation (3) we show that the sum of the marginal risk metrics leads to the total portfolio metric. Rearranging in Equation (4), we derive the marginal TVaR, which is also our component TVaR.
Effectively we have two metrics converging, and done with only one computation. To put it differently, once we have our stand-alone metrics and total portfolio TVaR, the marginal metric is analytical.
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