Scaling One-Day VaR to a 99.97% Ten-Day Estimate

Summary

The document examines a method for estimating a 99.97% ten-day value at risk from a 99% one-day historical simulation based on 500 days of observations. It describes scaling the one-day figure by the ratio of normal-distribution quantiles for the two confidence levels, then multiplying by the square root of the ten-day horizon. The example turns a $100 one-day VaR into $466 at the higher confidence level and longer horizon.

The response connects the 99.97% level with a one-year default probability of three basis points for AA-rated entities, suggesting this may explain why the figure appears in regulatory contexts. It characterizes the normal approximation as more suitable for broad securities portfolios than credit portfolios and potentially acceptable from a regulatory perspective. The exchange does not establish a specific regulatory rule or provide empirical validation of the scaling method. It also leaves the question of tail underestimation and alternative estimation methods largely unanswered, so the result should be read as an approximation rather than a demonstrated tail-risk estimate.

Key ideas

• A one-day historical VaR can be scaled to a longer horizon using the square root of time under stated assumptions.
• The proposed confidence-level adjustment uses a ratio of standard normal quantiles.
• The 99.97% level is associated in the response with a one-year default probability of three basis points for AA-rated entities.
• The response views normal scaling as more defensible for securities portfolios than credit portfolios.
• The exchange does not quantify the method’s tail error or compare it with alternatives.

Tags

# 99.97% Percentile VaR Approximation


# 99.97% Percentile VaR Approximation












I have been working with a group which references a 99.97% 10-day VaR figure.

They calculate this value via a 99% 1-day historical simulation over 500 days and then scale it under the assumption of a normal distribution, i.e. $scale = \frac{\Phi^{-1}(0.9997)}{\Phi^{-1}(0.99)}$, and also accounting for time, $scale2 = \sqrt{10}$.

So a \$100, 99% 1-day VaR becomes a \$466, 99.97% 10-day VaR.

I have two questions:

What is the significance (possibly in regulatory capital requirements) of the 99.97% confidence level? A google search has this figure appearing too much for it to be an arbitrarily chosen value.

Is this method quite poor or standard practice? While there might not be enough data to bootstrap the higher confidence level, is the assumption of normality particularly weak in the tails giving rise to significant underestimate?

Bonus points for considered alternatives..

## Answer by Mats Lind (score 6, accepted)

https://quant.stackexchange.com/a/50245

The 99.97% confidence is somtimes referred to as corresponding to the 1-year probability of default of 3 bps for AA-rated entities. (Here for example https://papers.ssrn.com/sol3/papers.cfm?abstract_id=963233 ) The normal approximation works better for general securities portfolios than for credit portfolios and might thus be seen as good enough from a regulatory perspective.

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