Log Normal Distribution

What is Log-Normal Distribution?

A log-normal distribution is a continuous distribution of random variables whose logarithms are distributed normally. In other words, the lognormal distribution is generated by the function of ex, where x (random variable) is supposed to be normally distributed. In the natural logarithm of ex is the x, the logarithms of lognormally distributed random variables are normally distributed.

A variable X is normally distributed if Y = ln(X), where ln is the natural logarithm.

  • Y= ex
  • Let’s assume a natural logarithm on both sides.
  • lnY = ln ex which results into lnY = x

Therefore, we can say, if X being a random variable has a normal distributionNormal DistributionNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more, then Y has a lognormal distribution.

Log Normal Distribution

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Log-Normal Distribution Formula

The formula for the probability density function of the lognormal distribution is defined by the mean μ and standard deviation σ, which is denoted by:

Log-Normal Distribution Formula

Parameters of Log-Normal Distribution

The log-normal distribution is characterized by the following three parameters:

  • σ, the standard deviation of the log of the distribution, which is also called the shape parameter. The shape parameter generally affects the overall shape of the lognormal distribution, but it does not impact the location and height of the graph.
  • m, the median of the distribution, also known as the scale parameter.
  • Θ, the location parameter which is used to locate the graph on the x-axis.

The mean and standard deviation are two major parameters of the lognormal distribution, and it is explicitly defined by these two parameters.

lognormal-distribution

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The following figure illustrates the normal distribution and log-normal distribution.

From the above figure, we could note the following features of the log-normal distribution.

  • The log-normal distributions are positively skewed to the right due to lower mean values and higher variance in the random variables in considerations.
  • The lognormal distribution is always bounded from below by 0 as it helps in modeling the asset prices, which are not expected to carry negative values.
  • The lognormal distribution is skewed positively with a large number of small values and includes a few major values, which result in mean being greater than mode very often.

From the above figure, we could observe that log-normal distribution is bounded by 0, and it is positively skewed to the right, which could be noticed by its long tail towards the right. These two observations are considered to be the major properties of lognormal distributions. In practice, lognormal distributions proved very helpful in the distribution of equity or asset prices, while normal distribution is very useful in estimating the asset’s expected returns over a period of time.

Examples of Log-Normal Distribution

The following are some examples where log-normal distributions can be used:

  • The volume of gas in energy and petroleum reserve.
  • The volume of milk production.
  • The quantity of rainfall.
  • The potential lives of manufacturing and industrial units whose chances for survival are characterized by the rate of stress.
  • The extent of periods to which any infectious disease exists.

Application and Uses of Log-Normal Distribution

The following are applications and uses of the log-normal distribution.

  • The most commonly used and popular distribution is a normal distribution, which is normally distributed and symmetrical and forms a bell-shaped curve that has modeled various natural from simple to very complex.
  • But there instances where normal distribution face constraints where lognormal distribution can be easily applied. The normal distribution can consider a negative random variable,s but lognormal distribution envisages only positive random variables.
  • One of the various application where lognormal distribution is used in finance where it is applied in the analysis of assets prices. The expected return on assets is graphed in a normal distribution, but the prices of the assets are graphed in a lognormal distribution.
  • With the help of the lognormal distribution curve, we can easily calculate the compound rate of return on assets over a period of time.
  • In case we applied a normal distribution to calculate assets prices over a period of time, there are possibilities of getting returns less than -100%, which subsequently assumes the prices of assets less than 0. But if we use lognormal distribution to estimate the compound rate of returnRate Of ReturnRate of Return (ROR) refers to the expected return on investment (gain or loss) & it is expressed as a percentage. You can calculate this by, ROR = {(Current Investment Value – Original Investment Value)/Original Investment Value} * 100read more over a period of time, we can easily ward off the situation of getting negative returns as lognormal distribution consider only positive random variables.
  • A price relative is the asset’s price at the end of the period divided by the initial price of the asset, which is equal to 1 plus holding period returns. To find the end of the asset of the period price, we can get the same by multiplying it with relative price times the initial assets price. Lognormal distribution takes only positive value; therefore, the asset price at the end of the period cannot be below 0.

Log-Normal Distribution in Modelling Equity Stock Prices

The log-normal distribution has been used for modeling the probability distributionProbability DistributionProbability distribution is the calculation that shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required. It is a mathematical function that gives results as per the possible events.read more of stock and many other asset prices. For instance, we have observed lognormal being appears in the Black-Scholes-Merton option pricing model, where there is an assumption that the price of an underlying asset option is lognormally distributed at the same time.

Conclusion

  • The normal distribution is the probability distribution, which is said to be the asymmetrical and bell-shaped curve. In a normal distribution, 69% of the outcome falls within one standard deviation, and 95% falls within the two standard deviations.
  • Due to the popularity of normal distribution, most people are familiar with the concept and application of normal distribution, but at the time, they don’t seem equally familiar with the concept of the lognormal distribution. The normal distribution can be converted into lognormal distribution with the help of logarithms, which becomes the fundamental basis as the lognormal distributions consider the only random variable which is normally distributed.
  • Lognormal distributions can be used in conjunction with the normal distribution. Lognormal distributions are the outcome of assuming the ln, natural logarithm in which base is equal to e = 2.718. In addition to the given base, the lognormal distribution could be made using another base, which would subsequently impact the shape of the lognormal distribution.
  • The lognormal distribution graphs the log of normally distributed random variables from the normal distribution curves. The ln, the natural log is known e, exponent to which a base should be raised to get the desired random variable x, which could be found on the normal distribution curve.

This has been a guide to what is log-normal distribution and its definition. Here we discuss examples of log-normal distribution along with its parameters, applications, and uses. You can learn more about finance from the following articles –

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