A log-normal distribution is a statistical distribution of logarithmic values derived from a related normal distribution. It can be translated to a normal distribution and vice versa using associated logarithmic calculations.
Understanding Normal and Log-Normal Distributions
A normal distribution is a probability distribution of outcomes that are symmetrical and form a bell curve. In a normal distribution, 68% of the results fall within one standard deviation, and 95% fall within two standard deviations.
While most people are familiar with a normal distribution, a log-normal distribution might be less well-known. A normal distribution can be transformed into a log-normal distribution using logarithmic mathematics since log-normal distributions originate from normally distributed sets of random variables.
There are several reasons to use log-normal distributions alongside normal distributions. Most log-normal distributions come from the natural log, where the base is equal to e (approximately 2.718). However, the distribution’s shape can vary if another base is used. Ultimately, the log-normal distribution plots the logarithm of random variables from a normal distribution curve.
Applications and Uses of Log-Normal Distribution in Finance
Normal distributions can sometimes allow for negative random variables, while log-normal distributions are all positive. This makes log-normal distributions particularly useful for analyzing stock prices.
One common application in finance is analyzing stock returns. While potential stock returns can be graphed in a normal distribution, stock prices are more accurately represented using a log-normal distribution. This can help identify the compound return that stocks may achieve over time.
Log-normal distributions are positively skewed with long right tails due to low mean values and high variances in random variables.
Log-Normal Distribution in Excel
You can calculate the log-normal distribution in Excel using the LOGNORM.DIST function:
LOGNORM.DIST(x,mean,standard_dev,cumulative)
Where:
x is the value at which to evaluate the functionMean is the mean of ln(x)Standard Deviation is the standard deviation of ln(x), which must be positive
Related Terms: normal distribution, stock analysis, compound return, logarithmic calculations, statistical distributions.
References
- Microsoft. “LOGNORM.DIST function”.
Get ready to put your knowledge to the test with this intriguing quiz!
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## What is a log-normal distribution?
- [ ] A normal distribution
- [ ] A distribution observed primarily in logarithms
- [x] A probability distribution of a random variable whose logarithm is normally distributed
- [ ] A distribution only used in tech analysis
## Which of the following describes a key characteristic of a log-normal distribution?
- [x] It is positively skewed
- [ ] It is symmetrical around the mean
- [ ] It can take negative values
- [ ] It has mean and variance equal to zero
## In finance, which of these is commonly modeled with a log-normal distribution?
- [ ] Annual returns on a portfolio
- [ ] Interest rates
- [x] Stock prices
- [ ] Profit margins
## Why is log-normality an important assumption in financial modeling?
- [ ] Because prices are always constant
- [x] Because it ensures prices cannot go below zero and can take on a wide range of values
- [ ] Because it simplifies risk calculations
- [ ] Because it renders symmetrical projections
## What properties do log-normal distributions share with normal distributions?
- [ ] They both are symmetrical
- [x] They are both used to model random variables
- [ ] They both can assume negative values
- [ ] They have identical mean values
## In a log-normal distribution, what shape does the frequency distribution of values take?
- [ ] Bell-shaped
- [ ] Uniform
- [x] Right-skewed
- [ ] Left-skewed
## Which statistical measure is particularly higher for log-normal distributions compared to normal distributions?
- [ ] Median
- [ ] Standard deviation
- [x] Skewness
- [ ] Variance
## What would happen to the distribution if the variable is log-normal and we apply a logarithmic transformation?
- [ ] It would become bimodal
- [ ] It would flatten out
- [x] It would become approximately normal
- [ ] It would reflect a left-skew
## To estimate future financial values, under what assumption is log-normal distribution used?
- [x] Under the assumption of exponential growth
- [ ] Under the assumption of mean reversion
- [ ] Under the assumption of uniform distributions of error terms
- [ ] Under the assumption of bond price variations
## Which of these transformation techniques turns a dataset of log-normal distributed variables into a set of normally distributed variables?
- [ ] Standardization
- [ ] Min-max scaling
- [x] Taking the natural logarithm of all variables
- [ ] Normalization
