Historical simulation refuses to assume anything and pays for that refusal with total dependence on whatever happened to be in your sample. The parametric method (also called variance-covariance or delta-normal) goes the opposite way: assume returns are Normally distributed with mean mu and standard deviation sigma (both estimated from data, or supplied as a risk-manager's forward-looking view), and read the answer straight off the Normal distribution's own formula, with no sorting, no sample noise, and no tail-observation scarcity at all.
How is parametric VaR calculated?
The mechanics lean entirely on two functions QM1.01 already built your intuition for: the quantile function (scipy.stats.norm.ppf, the inverse of the cumulative distribution, "what value has this much probability below it") and the density function (scipy.stats.norm.pdf, the height of the bell curve at a point). Let alpha = 1 - confidence be the left-tail probability and z = norm.ppf(alpha) (a negative number, at 95% confidence, z ≈ -1.6449; at 99%, z ≈ -2.3263). Then:
- VaR
= -(mu + z * sigma), the Normal distribution's ownalpha-quantile, sign-flipped to a positive loss. - ES
= -mu + sigma * phi(z) / alpha, the closed-form expectation of the Normal distribution's own left tail beyond that quantile, wherephiis the standard Normal density.
Take a dollar-denominated, S&P-linked book with mu = 0.03% and sigma = 1.1% per day (again, illustrative parameters, not a live position) at two confidence levels:
| confidence | z | VaR | ES | ES / VaR |
|---|---|---|---|---|
| 95% | −1.6449 | 1.779% | 2.239% | 1.258 |
| 99% | −2.3263 | 2.529% | 2.902% | 1.147 |
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How do VaR and ES change with the confidence level?
Two patterns are worth internalising, because both generalise far beyond this one example. First, VaR rises with confidence: 99% VaR is always at least as large as 95% VaR on identical data, because you are reading a rarer, farther-out quantile. Second, the ES/VaR ratio shrinks as confidence rises, under the Normal assumption, ES/VaR is 1.369 at 90% confidence but only 1.123 at 99.5%, because at very high confidence the tail you are averaging over is so thin it barely differs from its own boundary. This is precisely the parametric-method mirror of the "VaR = ES when k = 1" coincidence you saw in historical simulation, different mechanism, same underlying truth: a razor-thin tail sample (however it was produced) makes VaR and ES converge. Code pack exercise QD2.02-e02 implements exactly these two formulas and grades them against an India-flavoured and a US-flavoured parameter set, checking both the closed-form values themselves and the monotonicity properties (VaR rises with confidence; ES always exceeds VaR) that must hold for any correct implementation, not merely the one worked example above.
Why can parametric VaR understate risk?
The parametric method's entire value and entire danger live in one sentence: it is exact under its own assumption, and it says nothing about how well that assumption fits reality. Real daily returns are well known to have fatter tails than a Normal distribution predicts, large moves happen more often than the bell curve says they should (a fact with its own name, excess kurtosis, that QM1.01 covers as a distributional property; here you simply need to know that it means the parametric method systematically understates true tail risk on real return data). A parametric VaR is a mathematically exact answer to a question about a world slightly less scary than the one you actually trade in.