VaR That Understates the Tail
A trading firm reports 1-day 99% VaR of $4.2 million using a historical simulation with 250 days of data. A new risk analyst notices that the model has produced only three exceptions in the last 12 months against an expected 2.5, which looks acceptable - until she realizes the 250-day window still contains the calm pre-crisis period and excludes the most volatile month because of a data gap. Back-testing on a reconstructed series shows 7 exceptions in the same period. The desk's largest position is a concentrated emerging-market bond that trades thinly and has no reliable daily price; it is marked using a broker quote that has not changed in 11 days. The CRO wants a defensible number for the next board pack.
1. Explain why the reported VaR is likely understated and quantify the direction of the bias.
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The reported $4.2 million is understated for three compounding reasons. First, the historical window is stale and unrepresentative: it includes a calm regime and excludes the most volatile month due to a data gap, so the simulated distribution is truncated on the loss side. Reconstructed back-testing shows 7 exceptions versus 3 on the flawed data - at 99% confidence over ~250 days, the expected count is about 2.5, so 7 is a meaningful clustering signal, not bad luck. Second, the concentrated EM bond is illiquid and priced off a stale broker quote unchanged for 11 days; historical simulation cannot capture moves that never appear in the price series, and marking to a stale quote suppresses observed volatility. Third, concentration itself is not captured by a VaR that treats the position as diversifiable. The direction of bias is unambiguously downward - the true tail loss is larger than $4.2 million, and the board should be told the number is a floor, not a point estimate.
2. Recommend how to produce a defensible number for the board, including treatment of the illiquid position.
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A defensible number requires fixing the inputs and disclosing the method's limits. Steps: (1) rebuild the historical window to include the volatile period and repair the data gap; (2) supplement historical simulation with a stressed VaR or expected shortfall so the tail beyond the 99% point is visible; (3) apply a liquidity-adjusted or proxy-based treatment to the EM bond - use a comparable liquid index as a proxy, widen the price with a valuation uncertainty reserve, and stress it with a haircut reflecting the 11-day stale quote; (4) present a range (e.g., base VaR plus stressed and liquidity-adjusted figures) rather than a single point, with the assumptions stated; (5) document the back-testing exception analysis and the model limitation in the board pack. The board should see both the headline number and the caveat that thin markets make any single VaR unreliable. Presenting a range with clear assumptions is more defensible than a precise figure built on stale data.