When should I use lognormal VaR instead of normal VaR?

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AcadiFi · Accepted Answer

Lognormal VaR models the price level as lognormal, equivalently log returns as normal, which enforces positive prices and a skewed loss distribution. The VaR on price level V at confidence c is V * (1 - exp(mu - z * sigma)) where mu and sigma are the mean and standard deviation of log returns over the horizon. For short horizons the lognormal and normal VaR nearly coincide because exp(x) is approximately 1 + x for small x; for long horizons they diverge because the lognormal's right tail stretches and left tail compresses. Example: V = $100, horizon sigma = 20%, mu = 0, confidence 99%. Normal VaR = 2.326 * 0.20 * 100 = $46.52. Lognormal VaR = 100 * (1 - exp(-2.326 * 0.20)) = 100 * (1 - 0.627) = $37.30. The lognormal answer is lower because the asset cannot lose more than its current value. For leveraged portfolios, lognormal is inappropriate because losses can exceed principal; use normal or a specialized model. For positions where the natural log is tractable — equities, commodities, FX spot — lognormal is the cleaner choice for horizons beyond a few weeks. For FRM questions, remember that the two give similar answers short-term, that lognormal bounds loss at 100% of notional, and that the quantile on log returns is the normal quantile, not a separate distribution lookup.

When should I use lognormal VaR instead of normal VaR?

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