Log-normal distribution

Definition

A log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. This means that if the variable itself is not normally distributed but can be transformed by taking the logarithm, the result will follow a normal distribution. It’s especially useful in fields like finance and environmental studies, where data often cannot be negative and can be skewed.

5 Must Know Facts For Your Next Test

1. Log-normal distributions are commonly used to model stock prices and income distributions since they can only take positive values and can be highly skewed.
2. In a log-normal distribution, the mean, median, and mode are all different, with the mean being greater than the median and median greater than the mode.
3. The parameters of a log-normal distribution are typically defined in terms of the underlying normal distribution from which it is derived.
4. Log-normal distributions can be characterized by their scale parameter and shape parameter, which determine how spread out the distribution is.
5. In the context of pension funds, log-normal distributions can model asset returns or liabilities that evolve over time due to various economic factors.

Review Questions

• How does a log-normal distribution differ from a normal distribution in terms of its characteristics and applications?
  • A log-normal distribution differs from a normal distribution primarily because it can only take positive values and is often right-skewed, while a normal distribution can take any real number value and is symmetric. In applications, log-normal distributions are frequently used to model scenarios like stock prices or income levels, where values cannot be negative and tend to cluster towards one side. This makes them particularly useful in financial contexts, unlike normal distributions which may apply more broadly.
• What role does geometric Brownian motion play in understanding log-normal distributions within financial modeling?
  • Geometric Brownian motion is fundamental in finance for modeling asset prices because it assumes that the logarithm of price returns follows a normal distribution. This aligns perfectly with the characteristics of log-normal distributions, where asset prices themselves are positively skewed. The use of geometric Brownian motion ensures that modeled stock prices remain positive over time, reinforcing how log-normal distributions apply to real-world financial scenarios.
• Evaluate the implications of using a log-normal distribution for modeling pension fund asset returns compared to other distributions.
  • Using a log-normal distribution for modeling pension fund asset returns has significant implications for risk assessment and financial planning. Unlike other distributions that may not account for the skewness inherent in financial returns, a log-normal model captures the reality that most returns will be positive while allowing for occasional large gains. This characteristic helps actuaries better estimate future liabilities and ensure that funds are adequate for future payouts. Consequently, applying this model enables more accurate forecasting and strategic decision-making regarding investments within pension funds.

Related terms

Normal Distribution: A probability distribution that is symmetric around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Stochastic Process: A mathematical object defined as a collection of random variables representing the evolution of a system over time.

Geometric Brownian Motion: A stochastic process used to model stock prices in finance, where the logarithm of the price follows a normal distribution.