Intuitively Predict Insurance Chaos

A stochastic description of insurance loss ratios: signal and noise.

What’s in this post?
Building intuition of losses predictability with empirical observation for insurance portfolios clustered using lines of business.


It is commonly agreed upon, in fields such as probability theory, that random processes are mathematical objects. Historically, random variables were viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.

Volumes of earned premiums for Commercial Auto Liability insurance business underwritten in the US

What is a Line of Business?

The insurance industry usually refers to a standardized classification of lines of business such as fire, commercial, personal, auto, or residential. Within a portfolio and for a given line of business, insured risk profiles are very heterogeneous. However, the exposures share fundamental characteristics and lines of business are de facto arbitrarily clusters of insureds toward portfolios.
We can, for the sake of this post, look at commercial auto liability insurance in the US. We can easily access (public) historical data from annual financial reports up to 2017.

Annual US industry Loss Ratio distributions for Commercial Auto Liability between 1987 and 2017

How random are Insurance Losses?

Predicting if an accident (and insurance claim follow-ups) will occur for an individual account is impossible. 
What is possible however is to associate a likelihood to it. For example, we can arbitrarily say that it follows a Bernoulli process. And, after making assumptions regarding many similar accounts being independent and identically distributed for many years, fit a numerical probability P to it.

Predicting the exact financial losses for a given portfolio of insureds is also extremely challenging, nearly impossible. Nevertheless, what is possible is to predict its statistical behavior as a random variable. This is a lot like predicting the short term behavior of chaotic dynamical system with uncertainty and that means using a stochastic approach.

Histogram of ALL historical industry Loss Ratios for Commercial Auto Liability

This becomes even more intuitive when we look at the entire industry for a given line of business. Assuming that different insurance companies for different accident year are sampled from a distribution and are independent and identically distributed, it is very tempting to fit a Gaussian distribution to it and start computing its mean and standard deviation. It is also tempting to observe the third statistical moment and fit more skewed distributions such as gamma or log-normal distribution.

Volume reduces uncertainty

Modern portfolio theory argues that an investment’s risk and return characteristics should not be viewed alone, but should be evaluated by how the investment affects the overall portfolio’s risk and return. Information theory explains high predictability order at large scale and statistical thermodynamics illustrate how chaotic local phenomenon contribute to highly predictable motions at scale.
In finance, diversification is the process of allocating capital in a way that reduces the exposure to any one particular asset or risk. A common path towards diversification is to reduce risk or volatility by investing in a variety of assets. A parallel can be drawn with the volatility of insurance loss.

All historical industry Loss Ratios for Commercial Auto Liability from 1987 to 2017 (Y Axis) plotted vs volumes in earned premiums (X Axis)

More volume typically means less volatility in the Loss Ratio, and therefore less variability in the performance of an insurance portfolio.

For the sake of building an illustrative example, we can fit a probability distribution function and corresponding cumulative distribution function to historical loss ratios in this Lines of business. We can also filter out for a specific threshold of earned premium.


The study of complex systems in other fields benefits to the insurance industry for the measure of risk. Lines of Business reporting and intuitive statistics allow for elementary understanding of risk and this contributes to making the industry more transparent. A positive outcome is the possibility of more informed insurance investing.