# Lognormal Var Formula Question

Hello, I’m confused about a very easy question, how to calculate the lognormal Var. Here you are the question and answer:

The annual mean and volatility of a portfolio are 12% and 30%, respectively. The current value of the portfolio
is GBP 2,500,000. How does the 1-year 95% VaR that is calculated using a normal distribution assumption
(normal VaR) compare with the 1-year 95% VaR that is calculated using the lognormal distribution assumption
(lognormal VaR)?
a. Lognormal VaR is greater than normal VaR by GBP 487,050
b. Lognormal VaR is greater than normal VaR by GBP 787,050
c. Lognormal VaR is less than normal VaR by GBP 487,050
d. Lognormal VaR is less than normal VaR by GBP 787,050
Explanation: Normal VaR is calculated as follows:
Normal VaR (%) = Rp – z = 0.12 – (1.645 * 0.3) = 0.3735 = 37.35% (dropping negative sign)
and, Lognormal VaR is calculated as follows:
Lognormal VaR (%) = 0.12 – e[Rp – z] = 0.12 – exp [0.12 – (1.645 * 0.3)] = 0.56832 = 56.83%
Hence, Lognormal VaR is larger than Normal VaR by: 56.83% – 37.35% = 19.48% per year. With a portfolio
of GBP 2,500,000 this translates to VaR = 0.1948 x GBP 2,500,000 = GBP 487,050.

As I understood, the formula should be

Lognormal VaR (%) = 1 – e[Rp – z] , not Rp-– e[Rp – z] (used in this case)

Best Regards

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There was an errata to this question:

The annual mean and volatility of a portfolio are 12% and 30%, respectively. The current value of the portfolio is GBP 2,500,000. How does the 1-year 95% VaR that is calculated using a normal distribution assumption (normal VaR) compare with the 1-year 95% VaR that is calculated using the lognormal distribution assumption (lognormal VaR)?
a. Lognormal VaR is greater than normal VaR by GBP 487,050
b. Lognormal VaR is greater than normal VaR by GBP 154,500
c. Lognormal VaR is less than normal VaR by GBP 487,050
d. Lognormal VaR is less than normal VaR by GBP 154,500

Answer was “d” using the usual method to calculate lognormal VaR

thx, that explained

Thanks Prasad.Kelkar, just stumbled over that error too and was clarified thanks to your post

Normal: (R - (sigma) (zeta)) * VP

Log Normal: (1 - exp (R - (sigma) (zeta))VP

One should pay attention if asked to recalculate return and vol. for one period to another if required in question (like an annual to 1 day VaR f.ex.) or change conf. level (zeta) VaR * (2,328 / 1,645) to recalculate 95 % VaR to 99 % VaR which is a common case.

Cheers!

play it again, Sam