Back to Community
Log-normal Distributions for Returns

Hi Folks,

Log-normal distributions are very common in quant finance. They're used as many believe they more closely model stock behavior than other distributions. Some folks have asked how to take log prices and I realized we didn't have any materials on that. I decided to write a quick example notebook showing how to take log returns, and threw in some log-normality testing while we're in there.

The notebook includes a computation to check what percentage of your universe is log-normally distributed at any point in time.

You can always find more hands-on quant finance content in our free lectures:
https://www.quantopian.com/lectures

If you'd like to learn more about the Quantopian platform and APIs, check out our getting started tutorial:
https://www.quantopian.com/tutorials/getting-started

Loading notebook preview...
Notebook previews are currently unavailable.
Disclaimer

The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by Quantopian. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. No information contained herein should be regarded as a suggestion to engage in or refrain from any investment-related course of action as none of Quantopian nor any of its affiliates is undertaking to provide investment advice, act as an adviser to any plan or entity subject to the Employee Retirement Income Security Act of 1974, as amended, individual retirement account or individual retirement annuity, or give advice in a fiduciary capacity with respect to the materials presented herein. If you are an individual retirement or other investor, contact your financial advisor or other fiduciary unrelated to Quantopian about whether any given investment idea, strategy, product or service described herein may be appropriate for your circumstances. All investments involve risk, including loss of principal. Quantopian makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances.

3 responses

It may be obvious maths, but worth pointing out that log returns are just the difference in consecutive log prices.

At an intuitive level, log returns make sense. Prices represent a NPV of a future dividend stream, so changes to the growth rate or the discount rate in the NPV calculation will have an exponential effect on price. However, changes to the near term prospects of the company, such as reduced profitablility due to margin pressure from competition, have only a linear effect on the NPV calculation, which is a sub-exponential effect, clearly. I believe this causes the pronounced peak in the observed distribution of log returns.

@Delaney, Thank You very much for above code. This is exactly what I was looking for. Could you please advise how you would change the code in the last line so it will record a list of all those securities which are normally distributed? In below code, instead of getting percentage, I want to record the name (top row). Thank You for your help.

or i in range(num_assets):
# Get the series for the asset
log_R = data_output.iloc[:,i]
result = normaltest(log_R)
if result.pvalue >= significance_level:
num_normal += 1

There are several ways to get the list of securities which are normally distributed. Since the pipeline output is a pandas dataframe, I prefer using pandas methods rather than loops and lists. A few lines of code can do it.

# Assume log_returns is a pipeline output dataframe with our returns  
# The following filters for just those securities which have non-nan data for all days  
complete_groups = log_returns.dropna().groupby(level=1).filter(complete_data)

# Apply a function to each security to find the p-values. Group by level 1 which are the securities.  
p_values = complete_groups.groupby(level=1).daily_returns.agg(get_pvalue).to_frame('p_value')

# Finally compare these p-values to our significance_level to find all securities which are likely normally distributed  
normally distributed_securities = p_values.query('p_value >= @significance_level')

The resulting dataframe 'normally distributed_securities' are our securities which likely have returns which are normally distributed.
The index contains the security objects and the column is the associated p_value.

Check out the attached notebook. Several cells have been added to the bottom with the above code along with the associated helper functions and updated pipeline definition for log returns.

Loading notebook preview...
Notebook previews are currently unavailable.
Disclaimer

The material on this website is provided for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation or endorsement for any security or strategy, nor does it constitute an offer to provide investment advisory services by Quantopian. In addition, the material offers no opinion with respect to the suitability of any security or specific investment. No information contained herein should be regarded as a suggestion to engage in or refrain from any investment-related course of action as none of Quantopian nor any of its affiliates is undertaking to provide investment advice, act as an adviser to any plan or entity subject to the Employee Retirement Income Security Act of 1974, as amended, individual retirement account or individual retirement annuity, or give advice in a fiduciary capacity with respect to the materials presented herein. If you are an individual retirement or other investor, contact your financial advisor or other fiduciary unrelated to Quantopian about whether any given investment idea, strategy, product or service described herein may be appropriate for your circumstances. All investments involve risk, including loss of principal. Quantopian makes no guarantees as to the accuracy or completeness of the views expressed in the website. The views are subject to change, and may have become unreliable for various reasons, including changes in market conditions or economic circumstances.