< List of probability distributions

A **severity distribution** – also known as a *loss severity distribution* – predicts the range of losses that can arise from each operational loss event, instead of providing an exact value or set figure. This distribution family, which shows the variance or spread of potential losses [1], has extensive applications in areas such as insurance, finance, and risk management.

## Types of severity distribution

Severity distributions are a broad family that includes these popular distributions:

- Exponential distribution: characterized by a constant hazard rate and used for modeling time between events
- Weibull distribution: characterized by two parameters, shape and scale. It models a broad range of phenomena
- Gamma distribution: also characterized by shape and scale parameters, it models accident severity and various other events. However, if your data is very heavy-tailed, the Pareto distribution may be a better choice [2].
- Lognormal distribution: a skewed distribution used for modeling the severity of natural disasters.
- G-and-H distribution: Developed by Tukey in 1977, it is seldom used perhaps because it does not have a closed-form solution for a probability density function (PDF) [3].

When using a normal distribution for a severity distribution, some issues arise. For one, the distribution centers at zero, leading to half the values being negative and the other half positive. To fit this shape, data with a mean that is significantly higher than the standard deviation would work better, resulting in almost entirely positive data points. The distribution’s tails are short, leading to low value data points receiving insufficient weight [4].

The choice of the right severity distribution depends on the application. Opt for one that best describes the data and yields accurate predictions. To estimate a severity distribution, there are two ways: parametric, which assumes the general shape of a distribution, or non-parametric, which makes no assumptions about a fit to any distribution [4].

## History of the severity distribution

There is no single “inventor” of the severity distribution; it has been studied by a diverse group of scholars over the years. The term originated with Ladislaus Bortkiewicz, a Polish Mathematician, studied the distribution of accidents in factories in the late 1800s [5]. Bortkiewicz found that the distribution of accident severity was not normal, but instead followed a different distribution that he called the “severity distribution.”

Emil Julius Gumbel, a German mathematician, focused on the distribution of extreme values; his book *Statistics of Extremes* [6] was the first work devoted to extreme values. It is now considered a classic. The Weibull distribution, developed by English statistician Maurice Kendall, is another notable contribution. Over the years, the severity distribution has evolved into many varieties to model an array of phenomena.

Severity distributions find widespread applications across different fields:

- In insurance, severity distributions aid in deriving accurate cost estimations for insurance claims. For instance, using a severity distribution, an insurance company can assess the cost of a claim regarding a car accident.
- In finance, severity distributions are useful in estimating the risk of financial losses. They can come in handy when banks need to evaluate the chances of losses from loan defaults.
- Companies use severity distributions to develop robust risk management strategies. Through a severity distribution, they can create plans that minimize the impact of events like natural disasters.

## References

[1] Klaassen, P. & Eeghen, I. (2009). Economic Capital: How It Works, and What Every Manager Needs to Know 1st Edition. Elsevier Science

[2] Liu, S. & Forrest, J. (2010). Advances in Grey Systems Research (Understanding Complex Systems). Springer.

[3] Turley, P. Just a few more moments: the g-and-h distribution. Retrieved July 8, 2017 from: https://www.researchgate.net/publication/251947280_Just_a_few_more_moments_the_g-and-h_distribution

[4] Promislow, D. (2014). Fundamentals of Actuarial Mathematics. Wiley.

[5] L. von Bortkiewicz. 1898. Das Gesetz der kleinen Zahlen [The law of small numbers].

Leipzig: B.G. Teubner. The imprint lists the author as Dr. L. von Bortkewitsch.

[6] Emil Julius Gumbel. Statistics of Extremes. Dover Publications, Inc., 2004.