Weighted VaR

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Atoti Market Risk provides the “Weighted” variations of the VaR and ES measures which implement the WHS - exponentially weighted historical simulation approach.

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What is Weighted VaR?

Weighted VaR is a variation of the traditional Value at Risk (VaR) metric used in financial risk management. It estimates the potential loss in value of a portfolio over a defined period for a given confidence level, but with more emphasis on recent data.

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Key Concepts

• Traditional VaR assumes that all historical data points are equally important.
• Weighted VaR assigns greater weight to more recent observations, making it more responsive to current market conditions.

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Why use Weighted VaR?

• More responsive to recent market volatility.
• Better risk estimation during turbulent periods.
• Commonly used in RiskMetrics and Basel II/III frameworks.

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How is it calculated?

Weighted VaR is often calculated using Exponentially Weighted Moving Average (EWMA) or similar techniques. Here’s a simplified outline:

• Assign weights to historical returns, using an exponential decay factor (e.g., λ = 0.94).
• Calculate weighted standard deviation (volatility) of returns.
• Use this volatility to compute VaR at a given confidence level (e.g., 95% or 99%).

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Computation

The weight is computed for each PnL from the youngest to the oldest with the following formula: $w_i=\lambda^i\cdot \frac{\lambda-1}{\lambda^n-1}$ Where:

• $\lambda$ is the decay factor.
• $i$ is the index of the PnL for the scenario ordered from the youngest to the oldest starting with the index 0.
• $n$ is the number of scenario.

Please note that: $\sum_{i=0}^{n-1}w_i=1$ Then the PnL will be ordered from the best to the worst as it is done for the traditional VaR. Despite the traditional VaR, we will replace the quantile with the sum of the linked weights (from the best to the current one), and select the PnL that matches the confidence level.

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Configuration

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Input PnL vector

The simulated PL input data, which can be displayed using the PnLVectorExpand measure. The input order of the PnL vector (youngest to oldest or oldest to youngest) is configured by the property mr.var.weighted.pnl-oldest-first, by default the value is false that means that the youngest value comes first.

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Lambda

The default value for the decay parameter $\lambda$ is 0.94, it is configured with the confidence.defaults.weighted-var-lambda property and can be overridden using the WeightedVaRLambda context value.

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Rounding and quantiles

The way of selecting the sum of weight (that replaces the quantile) is always “CENTERED”, it means that we take the half of the current weight plus all the previous weights for the current ordered PnL. It is not configurable with the Quantiles hierarchy. $Q(PnL_i)=\frac{w(PnL_i)}{2} + \sum_{k=0}^{i-1}w(PnL_k)$ The applied rounding to compute the selected PnL uses the PnLs that surround the selected quantile and make an interpolation on the sum of weight. It corresponds to the “WEIGHTED” approach but, it is not configurable with the RoundingMethods hierarchy.

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Weighted ES

The Weighted ES (Expected Shortfall) uses the same sorting and centered quantile logic as the Weighted VaR, but instead of interpolating around the quantile boundary, it computes the weighted average of the PnL scenarios in the tail. The algorithm iterates through the PnLs sorted from worst to best, accumulating the weight and weighted PnL of each scenario. When the centered quantile $Q(PnL_i)$ first reaches the ES confidence level (controlled by ESConfidenceLevel), the Weighted ES is returned as: $\text{Weighted ES}=\frac{\sum_{k} w(PnL_k) \cdot PnL_k}{\sum_{k} w(PnL_k)}$ where the sum runs over all scenarios strictly before the one that crosses the quantile boundary.

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References

The implemented algorithm is described in Section 3 of the following paper:

• Richardson, Matthew P. and Boudoukh, Jacob and Whitelaw, Robert F., The Best of Both Worlds: A Hybrid Approach to Calculating Value at Risk (November 1997).

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See also

• WeightedVaRLambda