Tail Weighting Choices in Simulated Expected Shortfall Estimation

# Adding extra weight to the largest seen loss in ES/CVaR estimation


# Adding extra weight to the largest seen loss in ES/CVaR estimation












I'm working on estimating Expected Shortfall (ES) from simulated loss observations and came across a method that uses a specialized weighting scheme for the tail losses. In particular, for the case where the quantile rank $ k $ lies strictly between 1 and $ N $ (i.e., $ 1 < k < N $), the method applies the following weights:

- Setup:

Let:

- $ q $ be the loss quantile (e.g., $ q = 0.95 $ for the 95th percentile),

- $ N $ be the total number of loss samples (assumed sorted in increasing order),

- $ w $ be the basic weight defined as $ w = \frac{1}{N + 1}, $

- $ k $ be the rank corresponding to the quantile threshold, computed as $ k = \left\lfloor (1 - q)(N + 1) \right\rfloor, $

- and $ ac $ be an adjustment (or correction) factor

- Weighting Scheme for $ 1 < k < N $:

In the tail of the loss distribution, the method assigns weights as follows:

- Largest Loss Observation: The worst (largest) loss is given a weight of $ w_{\text{max}} = 1.5\,w. $

- VaR Observation (Loss at the Quantile Threshold): The loss corresponding to the Value-at-Risk (VaR) level is weighted by $ w_{\text{VaR}} = 0.5\,w + ac. $



- Computation of Expected Shortfall:

Finally, the Expected Shortfall is computed as the negative of the weighted average of the losses in the tail, scaled by the factor $ \frac{1}{1 - q} $: $ \text{ES} = -\frac{1}{1 - q} \sum_{i \in \text{tail}} w_i\, \ell_i, $ where each $ w_i $ is assigned as described above.

Question:

Is this approach of weighting the tail losses—specifically, assigning the largest loss a weight of $1.5w$ and the VaR observation a weight of $0.5w + ac$—a common or recommended method when estimating ES from simulations? Are there any potential pitfalls or improvements I should consider when using this scheme in risk management applications?

Any insights or references to literature that discuss similar weighting methods would be greatly appreciated.

## Answer by KaiSqDist (score 1)

https://quant.stackexchange.com/a/81877

Not a complete answer, but something related to what you are looking for is spectral risk measures.

I took this from page 263 of Risk Management and Financial Institutions by John C. Hull. As you can see, one condition to be a coherent risk measure is a monotonically increasing probability-weight for losses above the VaR level, which is fulfilled by all 3 variants below.

Your particular weighing schemes is only coherent IF $ac \leq 0.5w$, which makes the VaR loss weight lower or the same as the intermediate weights.

PS. this might not be as relevant, but https://www.sciencedirect.com/science/article/pii/S0304407622000380 talks about the equivalence of VaR and ES at different confidence levels and characterizing different empirical distributions. A very interesting paper on how the risk measures are related.

## Answer by lehalle (score 1)

https://quant.stackexchange.com/a/81886

It is natural to weight differently the occurrences that are 'below' your $1-q$ level. Indeed what you would lie to estimate

$${\rm ES}_q = \mathbb{E}(X \, \mathbf{1}_{X<\alpha_q})=\int_{X<\alpha_q} x\, d\mu_X(x).$$

If you discretize, usually you consider $$\int_{X<\alpha_q} x\, d\mu_X(x)\simeq \frac{1}{N q} \sum_{x_k<\alpha_q} x_k = \sum_{x_k<\alpha_q} \frac{1}{N q} x_k,$$

because you consider that this discretisation comes from the realisation of $p_k=\mathbb{P}(X=x_k)$, meaning that: *the uniform random variable $U_k\sim{\frak U}(p_k)$ gave a $1$ for your observation and a zero for what you do not observe.

Note that there is already weights that are $1/(qN)$.

But since they are very rare events, you could say that this ''$1$'' is not the correct estimate (you get the prior that it is rare by the way ; remember that the law of the estimation of a quantile $\tilde q$ is entered on the true value but has a variance of $\tilde q (1-\tilde q)/N$).

So you can decide to reweight arbitrarily your observations, deciding that a better estimator for $\int_{X<\alpha_q} x\, d\mu_X(x)$ is $ \sum_{x_k<\alpha_q} w_k x_k$.

For instance, it is know that the expectation of the minimum observed over $K$ realizations of a Uniform law is not its empirical minimum $x_\min$ for is $$\hat X_\min = x_\min - \frac{x_\max - x_\min}{K}.$$

What you do is similar to that.

I am nevertheless not aware of specific re-weighting schemes that would be 'standard'.