Is there a closed form solution for the quantiles of the lognormal distribution. And if so can they also be interpreted as Value at Risk measures? F.e. is the 5% quantile of a lognormal PDF equivalent to the 5% VaR?


1 Answer 1


Let $X$ be a normal r.v. and $Y = e^X$. The quantiles of $Y$ are simply obtained by exponentiating the quantiles of $X$.

Now there's no closed form expression for the quantiles of the normal distribution; consequently, there's no closed form expression for the quantiles of a lognormal.

However, as indicated in my opening paragraph, you can express the quantiles of a lognormal in terms of the quantiles of the corresponding normal. If $x_q = \mu+\Phi^{-1}(q)\cdot \sigma$ is the $q$-quantile of a normal with parameters $\mu$ and $\sigma$, then $y_q = e^{x_q}$ is the corresponding quantile of the lognormal with the same parameters.

Since functions are typically available which give normal quantiles to quite high accuracy, you can make use of those in calculations involving lognormal quantiles.

A value at risk (VaR) is a quantile, so these considerations apply to value at risk.

  • $\begingroup$ so lets say if I have Log(S) is normally distributed with mean 8.91 and standard deviation 1.95 (where S is a stock). If I use the formula u stated it would be possible to determine the VaR as a quantile i.e. for 95% it would be with (S_0 =50) 50*exp(8.91-1.645*1.95) right? I also use the one sided z measure as VaR is only one sided. $\endgroup$
    – macro123
    May 18, 2020 at 1:40
  • $\begingroup$ Yes, VaR is one-sided (investment VaRs are lower tail quantiles, insurance VaRs are upper tail quantiles). I'm not clear on what $S_0$ is in your question but the remainder of the calculation looks right. However, beware -- if your mean and standard deviation are not population quantities but are instead estimated from a sample, this changes things. $\endgroup$
    – Glen_b
    May 18, 2020 at 1:56
  • $\begingroup$ are these also additive? since for the normal case there is a formula. $\endgroup$
    – macro123
    May 18, 2020 at 2:16
  • $\begingroup$ what do you mean by additive here? What's additive with what? $\endgroup$
    – Glen_b
    May 18, 2020 at 3:37
  • $\begingroup$ say if I have two stocks with a VaR can I also somehow extract the VaR of a portfolio lets say if I have 1 unit of 1 stock and 0.5 units of the other. Is it possible to weight the VaRs to get the VaR of the portfolio? $\endgroup$
    – macro123
    May 18, 2020 at 3:56

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