# Thread: Stock market returns, Normal and Log-normal distributions - question

1. ## Stock market returns, Normal and Log-normal distributions - question

- I'm looking at expected stock market returns for a hypothetical portfolio
- We are assuming this portfolio has annual returns that are Normally distributed with a mean annual growth rate of 4.4% and the standard deviation of the annual growth rate is 13.1% (before anyone asks, I understand the limitations of assuming Normal returns on stock market returns with fat tails etc, but this is the assumption we are using).
- I don't think it matters, but to be clear, the 4.4% is a real (after inflation) return i.e. assuming 2.5% inflation, the actual return would be expected to be 6.9%.
- A software provider we use takes this portfolio and, using a log-normal distribution assumption, and some stochastic/Monte Carlo modelling, produce an "expected value of the portfolio after 1 year". As I understand it, they use log-normal as the portfolio cannot loose more than 100% of its value, and this would be a possibility if they used the Normal distribution.
- The figure they then quote as being the Median real return from this modelling is 3.5%.

My query is that this 3.5% seems a lot lower than the 4.4%. I just don't believe the difference, even if one is the Mean and the other the Median, and clearly nearly a 1% pa difference in "expected growth" makes a big difference to what someone's investment might look like in 30-40 years if looking at a pension pot.

I'm wondering if someone can help with the following (or provide some different comment if I am looking at it wrongly)

- Can we say what the parameters of a log-normal distribution would be that would most closely approximate to the returns of the Normal distribution quoted?
- Can we then use those parameters to work out what the Median of that log-normal distribution would be?
- Does that give a figure similar to 3.5%, or is my gut feeling correct and it is closer to the 4.4% of the Normal distribution mean?

The software provider is saying that the log-normal Median will always be lower than the Normal distribution Mean. I can accept that (I think!), but it is the scale of the difference that just doesn't feel right!

Many thanks for any help.

Norm
(embarrassed to be asking for help, but then my maths/stats degree is 25 years old…..!)

2. ## Re: Stock market returns, Normal and Log-normal distributions - question

OK, I've been working on this and made some progress, to the point where I think I can ask a simpler question.

Say you're looking at annual investment returns, which you are happy to assume are Normally distributed. If you were going to do a Monte Carlo type simulation using a log-normal distribution, what would you input as the mean of your log-normal distribution?

Would you use:

a) 1+x where x is the mean of the Normal distribution e.g. if x = 4.5% then you'd assume 1.045
or
b) e^(x + (y^2)/2) where x is the mean and y the stan dev of the Normal distribution.

In essence, would you have the log-normal distribution having the same mean (option a) as the Normal distribution (where by "same" I mean where 1.045 = 4.5%), or would you have them having different means (option b)? Which is a better modelling assumption?

Many thanks

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