Definition
The Log Normal Distribution, in finance, refers to a statistical distribution of logarithmic values from a related normal distribution. It is often used to model stock prices, assets, or other values that are expected to grow exponentially. This kind of distribution does not include negative values and is skewed, unlike the normal distribution which is symmetrical.
Key Takeaways
- Log Normal Distribution is a statistical distribution of random variables that have a normally distributed logarithm. It is heavily utilized in finance for modeling stock prices, asset prices, and other values that can’t go into negatives.
- Unlike a normal distribution, a log normal distribution is skewed with a heavier tail on the right, indicating the higher probability of large variations. This characteristic makes it beneficial in quantifying real-world phenomena where values cannot be less than zero.
- The parameters of Log Normal Distribution are the mean and standard deviation of the variables’ logarithmic values, not the original variables. This feature allows it to model multiplicative effects and compound returns in finance.
Importance
The log normal distribution is a crucial concept in the finance world, especially for describing and modeling investment returns.
This concept is significant because it allows for variations in returns to be asymmetrical, meaning returns can potentially have larger gains than losses, the exact type of outcome most investors are pursuing.
It also doesn’t allow for negative values, which makes it ideal for pricing assets such as stocks and options, since their prices can’t fall below zero.
Moreover, real-world phenomena such as home prices, income distribution, population growth, and even the size of particles output by a grinder or crusher, follow a log-normal distribution.
Therefore, understanding and applying log-normal distribution could lead to more accurate analysis and predictions, helping investors or businesses to steer their financial decisions wisely.
Explanation
Log Normal Distribution is a key concept in finance particularly used in modeling price movement of underlying assets like stocks due to its specific characteristic where it only takes on positive values. This makes it especially applicable to financial parameters that cannot go below zero such as stock prices, property values or interest rates.
A key feature of the log normal distribution is that it is skewed to the right, capturing the reality of asset price movements where large upward surges are possible while declines are naturally limited to the asset’s price falling to zero. Through Log Normal Distribution, we can also approximate the future price of an asset by acknowledging that prices can’t drop below zero and that potentially the asset price could rise to infinity.
It is prevalent in the Black-Scholes model used in options pricing and Capital Asset Pricing Model (CAPM) used in portfolio management to describe returns. Essentially, by representing the potential variations in asset prices, log normal distribution becomes a vital component in predicting asset behaviors and pricing financial derivatives.
Examples of Log Normal Distribution
Stock Prices: In finance, the most common real-world application of log-normal distribution is in the modeling of asset prices, including stocks. The assumption is that over time, a stock’s price follows an exponential growth rate, and the fact that a stock price can’t go below zero makes log-normal distribution a suitable model. Fluctuations in stock prices often exhibit a skewness that makes the log-normal distribution more appropriate to use than the normal distribution.
Insurance Claims: In the insurance industry, the log-normal distribution is often used to model loss amounts. This is because loss amounts can be assumed to take on positive values only and because the losses often show a kind of skewness where there are many small losses and few large ones – a pattern which can be captured by a log-normal distribution.
Income Distribution: The distribution of income within a population is often modeled using a log-normal distribution. This is because the distribution of income is necessarily positive and is often skewed, with a small number of people having very high incomes. In this case, the log-normal distribution can provide a good fit for the observed pattern of income distribution.
FAQ: Log Normal Distribution
1. What is a log normal distribution?
A log normal distribution is a statistical distribution of logarithmic values from a related normal distribution. It is used in various statistical studies where values are positively skewed, not symmetric like a normal distribution.
2. How is log normal distribution different from normal distribution?
The key difference is that in a normal distribution, values are distributed symmetrically around the mean, while in a log normal distribution, values are positively skewed. This means excessive values are on the right, not on the left, which gives the distribution a characteristic skewed shape.
3. When is log normal distribution used in finance?
Log normal distribution is often used in finance for modeling stock prices, returns, compound interest, etc, as they can never take negative values and are always skewed to the positive side. This makes log normal distribution better suited for these variables than the normal distribution.
4. How is a log normal distribution characterized?
A log normal distribution is characterized by a parameter, mu, representing the mean of the logarithmic values, and a parameter, sigma, representing the standard deviation of these logarithmic values. These two parameters uniquely define a log normal distribution.
5. What is the calculus behind log normal distribution?
The calculus behind log normal distribution involves complex mathematical theories including calculus, integration and logarithmic transformations. The detailed explanation goes beyond the scope of this FAQ, but in essence it uses a logarithmic transformation to normalize a positively skewed distribution.
Related Entrepreneurship Terms
- Volatility
- Compound Returns
- Brownian Motion
- Geometric Brownian Motion
- Skewness and Kurtosis
Sources for More Information
- Investopedia: A comprehensive website dedicated to providing clear and concise information on a vast range of financial topics, including log normal distribution.
- Khan Academy: Excellent educational website that provides free, detailed, and well-explained lessons on a host of subjects, including mathematics, finance, and economics.
- Coursera: A website that offers online courses from top universities around the world. Financial topics like log normal distribution are covered in depth in some of these courses.
- JSTOR: A digital library that provides access to a wide range of academic resources, including scholarly journals, books, and primary sources. Researchers and academics contribute many resources related to financial mathematics.
