Value at Risk: Methodsintermediate

Parametric (Delta-Normal) VaR

The parametric, or delta-normal, method assumes returns are normally distributed and reads VaR straight off the normal curve. Set the mean to zero over a short horizon and VaR is a multiple of the standard deviation $\sigma$ : scale by the one-tailed critical value $z_c$ and by the position value $V$ . The standard $z$ -values are $z = -1.28$ at 90%, $z = -1.65$ at 95%, and $z = -2.33$ at 99%. The method is fast and needs only a volatility estimate, but it inherits every weakness of the normality assumption.

Try it yourself

Value at Risk — three ways

How much could this book lose on a bad day? VaR draws a line in the loss tail at confidence 95%. The parametric line assumes a normal curve, VaR = (z·σ·√h − μ·h)·V with z = 1.645 (one-tailed). The historical and Monte-Carlo methods read VaR off a simulated loss distribution instead. Under normality the three roughly agree; fat tails pull the historical figure deeper.

1-day 95% parametric VaRUS$3.29M

Parametric (normal) · largestUS$3.29M

Historical (fat tails)US$3.05M

Monte-Carlo (5,000 draws)US$3.27M

z (95%, one-tailed) 1.645Horizon σ√h 2.00%Largest VaR Parametric

Portfolio value VUS$100M

Daily volatility σ2.0%

Daily expected return μ+0.00%

Horizon h1 day

Confidence level c

The three estimates land near US$3.29M, but the historical figure of US$3.05M is about the same, since this sample happens not to bite past the Gaussian tail here. The largest of the three is Parametric.

Discuss: which method would you report to a regulator, and which to your own risk committee?

Why it matters

If you believe returns are normal, the entire loss distribution is pinned down by one number, its volatility. VaR is then just "how many standard deviations into the tail does my confidence level sit," times the dollar size of the position. That is why the method is so popular: one volatility input and a lookup of $z$ gives an answer in seconds. The price you pay is assuming a bell curve, which understates how often markets deliver large moves.

Formulas

Parametric VaR (percent of value, zero mean)

\mathrm{VaR}_c = -z_c\,\sigma\,V

With

z_c < 0

(e.g.

z = -1.65

at 95%) the product

-z_c\,\sigma

is positive, giving a positive loss.

\sigma

is the return volatility over the chosen horizon and

V

is the position value.

Critical z-values (one-tailed)

z_{90\%} = -1.28, \quad z_{95\%} = -1.65, \quad z_{99\%} = -2.33

These are left-tail standard-normal quantiles. A 99% VaR sits 2.33 standard deviations below the mean, so a higher confidence level uses a larger multiple.

Worked examples

Scenario

A US$100M portfolio has a daily return volatility of $\sigma = 2\%$ and an assumed zero mean. Compute the one-day 99% parametric VaR.

Solution

Apply $\mathrm{VaR} = -z_c\,\sigma\,V$ . Plugging in gives 2.33 times 0.02 times US$100M, which equals US$4.66M. The desk should expect a one-day loss of at least US$4.66M on about 1 day in 100. Switching to 95% replaces the 2.33 multiplier with 1.65, giving US$3.30M, a smaller number for the same portfolio because 95% reaches less far into the tail.

Scenario

For the same portfolio, the risk team instead estimates $\sigma = 2.275\%$ per day. Find the one-day 99% VaR.

Solution

Now the loss multiplier gives 2.33 times 0.02275 times US$100M, which is about US$5.3M. VaR scales linearly with both the volatility input and the position size, so a 14 percent rise in $\sigma$ raises VaR by the same 14 percent. This is the headline US$5.3M on US$100M figure: a 99% one-day loss threshold of about 5.3 percent of capital.

Common mistakes

✗Parametric VaR makes no distributional assumption. It assumes returns are normal; that single assumption is exactly what produces the clean $z\,\sigma\,V$ formula and also its main weakness.
✗The 99% z-value is 2.58. The 2.58 figure is two-tailed; VaR is one-tailed, so the correct 99% critical value is $z = -2.33$ .
✗Delta-normal VaR works well for options. The delta-normal approach is a linear approximation; for portfolios with strong convexity, such as options, it misstates risk and a full-valuation method is needed.
✗A bigger z-value means the portfolio is riskier. The z-value reflects only the chosen confidence level; the risk is carried by $\sigma$ and $V$ , not by the choice of $c$ .

Revision bullets

•Assumes normal returns; VaR is a multiple of volatility
•Zero-mean form: $\mathrm{VaR} = -z_c\,\sigma\,V$
•One-tailed z: $-1.28$ (90%), $-1.65$ (95%), $-2.33$ (99%)
•Fast and data-light; only needs a volatility estimate
•Linear (delta) approximation, poor for option convexity

Quick check

A US$100M portfolio has daily $\sigma = 2\%$ and zero mean. Its one-day 99% parametric VaR is closest to

Which one-tailed critical value belongs to a 95% parametric VaR?

Connected topics

Normal & Z-Scores VaR Definition VaR With Mean Valuation Methods

Sources

Jorion (2007), Ch. 5
Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.
Develops the delta-normal VaR formula and the one-tailed normal critical values.
Hull (2018), Ch. 14
Hull, J. C. Risk Management and Financial Institutions. 5th ed. Wiley, 2018. ISBN 978-1-119-44811-2.
Covers the model-building (parametric) approach and the linear delta approximation.

How to cite this page

Dr. Phil's Quant Lab. (2026). Parametric (Delta-Normal) VaR. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-parametric-var

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