# Lognormal/normal distribution in returns?

Hi Guys,

just did the CFAI Study Session 3, Reading 9 EOC question 20 which asked about wether asset prices and returns are normal/lognormal distributed.

There is no question about lognormal distributed asset prices because of the abscence of negative prices.

The answer says also that “A normal distribution is suitable for describing asset returns.” (I hope, citing that one sentence is allowed…). In my opinion this is not true. A normal distribution of returns allows returns to be under -100% which would lead to negative asset prices. Therefore, a lognormal distribution should be appropriate.

I also attended a risk management course at university where we learned that “asset returns are lognormal distributed and asset logreturns are normal distributed”.

What is your take on that?

My take on it is that CFA Institute advocates that returns are normally distributed and that prices are lognormally distributed.

And they write the exams.

Thanks for your response! And btw thanks for all your good posts at AF.

I took a further look into the CFAI material which states the following:

“In the following we show that if a stock’s continuously compounded return is normally distributed, then future stock price is necessarily lognormally distributed.”

Of course I don’t doubt that statement since it clearly refers to continuously compounded returns. However, the question doesn’t say “continuously compounded returns” but only “returns”. In my opinion they are not the same.

If the question said continuously compounded returns, I would definitely agree. Maybe I’m too rigorous here…

Or the author of the question is too sloppy.

On CFAI Curriculum you will also find that normal distribution is OFTEN a good aproximation of any returns (not just continously compounded returns), also it doesn’t mater if stock’s continuously compunded returns are normally distributed or not because stock price will be well described by the lognormal distribution in either case.

So the conclusion is :

Normal distribution is suitable for describing asset returns and lognormal distribution for asset prices

Also to be realistic how often do you see returns below -100 % (=> not to use normal distribution) and how often you don’t see negative returns below 0 (=> to prefer lognormal distribution which is bounded from below by 0 )?

Of course I know that normal distribution is often a good approximation for returns. An I also know that asset prices can be described by a lognormal distribution.

For asset prices and regular (not continuously compounded returns) the following equation holds:

S1 = S0 * (1+r).

When you assume normally distributed returns, the pure mathematical result is normal distributed stock prices.

When you assume lognormally distributed returns, the pure mathematical result is lognormally distributed stock prices.

For CCR:

S1 = S0 * e^®

When you assume normally distributed CCR, the pure mathematical result is lognormally distributed stock prices.

However, as S2000magician said, I think the author was too sloppy. I hope he won’t be at the exam questions.