Lognormal Distribution

[Yash $$$]

Hi,

Could anyone please explain me the meaning of Lognormal Distribution,
(in simple terms), and what is it actually used for?

I know my doubt is silly (i have googled it several times too!), but i have failed to understand the real meaning of it as well as the meaning of "Lognormal Var" and its calculation,

please help

Thanks

 

[ShaktiRathore]

A variable x is normally distributed then the logarithm values of this variable x is said to be lognormally distributed. This is lognormal distribution. For e.g. if returns of stock ln(St/St-1) are normally distributed then ln(St/St-1) ~N(0,1) this means that Stock price St are lognormally distributed and the corresponding returns are normally distributed.
Lognormal Var considers all the returns as lognormally distributed log rt~N(mean,variance)=> rt~e^N(mean,variance), so if for normally distributed return rt VaR is (-mean+sigma*z)*V then for lognormally distributed returns VaR is [1-e^(mean-sigma*z)]*V.

thanks

 

[Yash $$$]

Thanks for the explanation Shakti

Does it mean that returns (in %) are normally distributed, but Prices (in $) are lognormally distributed
looking at the Var formula, i feel the difference between normally distributed Var & Lognormally distributed Var, is much like the difference between Arithmetic mean & Geometric mean....is that right?

lastly, how does Lognormal distribution help, as the difference between normally distributed Var & Lognormally distributed Var is very slim & kinda similar is most cases.

Appreciate the help

 

[ShaktiRathore]

Yes returns (in %) are normally distributed, but Prices (in $) are lognormally distributed. Yes its analogous to it, the mean of normal distrbn is arithmetic mean while that of lognormal is geometric which is mean-.5*sigma^2< Arithmetic mean of a normal distribution.
Lognormal helps to plot the price levels which are always positive unlike returns which are negative and is useful for charting a distribution of a variable like the stock price. This can help in anal sing the var seeing the price distribution.
thaanks

 

[lighta]

Sorry for the simple questions.

Could you please explain why there is a negative in front of mean? Normal dis. VaR is (-mean+sigma*z)*V

Also what are relative VaR and absolute VaR?

Thanks

 

[ShaktiRathore]

hi
The (-mean+sigma*z)*V is the absolute Var we are finding whats the greatest possible loss possible for the portfolio given the expected return mean for the portfolio. Now since the greatest possible loss is sigma*z*V which is the relative var. But when we look in absolute terms the total loss for the portfolio as a whole is sigma*z*V minus the return for the portfolio. Because we are earning return on the portfolio at the same time there is loss possible on it so that total net possible loss is sigma*z*V minus the return for the portfolio. So absolute var in way measures the absolute net loss considering portfolio return as well as the greatest negative downside possible for the portfolio so absolute Var considers worst net loss possible for the portfolio whereas relative var do not consider return on the portfolio but only questions as to what is the worst downside possible for the portfoio ignoring the return of the portfolio.
for more visit this link, David explains in best way possible(thanks to him for this generous effort): http://forum.bionicturtle.com/threads/relative-vs-absolue-var.7246/#post-25830

thanks

 

[lighta]

Thanks so much for the very clear explanation.