Banks compute VaR daily — it is easier to estimate a one-day loss distribution from daily observations than a ten-day distribution from non-overlapping observations. But regulators require a 10-day 99% VaR for the market risk capital charge (Basel II/2.5), and FRTB has now moved to an ES at 97.5% over a stress period and multiple liquidity horizons. The question is how to move from the one-day quantity that statistical models produce to the horizon that capital rules require.
Two complementary problems arise here: (1) how to estimate VaR and ES from a finite simulated or historical sample, and (2) how to scale a short-horizon risk measure to a longer one. The standard answer to (2) is the square-root-of-time rule , which is exactly right for i.i.d. Gaussian returns. It is badly wrong for GARCH processes and for portfolios with options. Understanding when it works and when it breaks is core market-risk literacy.
Learning objectives. By the end of this note you should be able to:
- derive the empirical quantile and ES estimators for a simulated loss sample;
- state the EVT-based GPD estimator for the tail of the loss distribution;
- define the -period loss operator and derive the square-root-of-time scaling from i.i.d. Gaussian assumptions;
- explain why the rule fails for GARCH processes and non-linear books;
- describe the Monte Carlo approach to multi-period risk;
- state the Basel market risk capital formula and explain the regulatory motivation for the FRTB switch from VaR to ES.
In this lesson
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- Estimating risk measures from simulated data
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- The -period loss operator
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- Square-root-of-time scaling
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- Monte Carlo approach to multi-period risk
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- Regulatory capital and FRTB
Before you start
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