Value-at-Risk Forecasting Under Multi-Period Capital Requirements

Summary

The document formulates a problem in which a trader wants to choose Value-at-Risk forecasts to reduce regulatory capital charges over a future evaluation window. The stated charge sums, across successive days, the larger of a VaR-based term involving a rolling average and a term based on that day’s VaR. A multiplier depends on the recent count of returns that breach the forecast, linking today’s VaR choices to later capital calculations.

The key difficulty is that minimizing the charge for the next date alone may not minimize the total charge over the full horizon. This makes the objective path-dependent: a forecast affects both the current charge and future rolling averages, while breaches influence the multiplier. The document asks for an approach but supplies no proposed optimization, empirical evidence, or specific forecasting model. Any solution would need to state its forecast constraints and account for sequential dependence and the evaluation horizon; the source does not establish which approach performs best.

Key ideas

• The capital objective aggregates VaR-related charges across a future evaluation window.
• Each daily charge depends on both a rolling average of past VaR forecasts and the current forecast.
• A multiplier tied to recent VaR breaches makes the objective depend on forecast outcomes over time.
• Optimizing the next day’s charge alone need not optimize the full horizon’s total charge.
• The document identifies a path-dependent optimization problem but offers no solution or empirical comparison.

Tags

# Value-at-Risk Calculation with respect to the Capital Requirements


# Value-at-Risk Calculation with respect to the Capital Requirements












I want to calculate the Value-at-Risk at date $t$ in such a way that I minimize the capital requirements given as \begin{align} \text{CR}_{\,t+1\,:\,t+250} = \sum_{h=0}^{249}\max\left( -(3+k_{t})\overline{\text{VaR}}^{60}_{t+h}, -\text{VaR}_{t+h}\right), \end{align} in which $\overline{\text{VaR}}^{60}_{t} = 1/60\sum_{\tau=1}^{60} \text{VaR}_{t-\tau+1}$ and $k_{t} = f(\sum_{\tau=1}^{250} H_{t-\tau+1})$ where $H_t = 1_{\left\{r_t < \text{VaR}_t\right\}}$.

The problem is that that $\text{VaR}_t$ minimizing the $\text{CR}_{t+1}$ does not necessarily minimize $\text{CR}_{\,t+1\,:\,t+250}$. Any idea how we can approach this problem is appreciated.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.