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Chance and Probability as Applied to Captive Insurance

By Dugald Macleod
12 September 2007

“Probability is expectation founded upon partial knowledge. A perfect acquaintance with all the circumstances affecting the occurrence of an event would change expectation into certainty, and leave neither room nor demand for a theory of probabilities.”
George Boole

“...trying to reach the future through the past.....”
Paul Brady

The decision to set up an insurance captive is an entrepreneurial one. It involves replacing a contract whose terms are known, with a business proposition, whose outcome is unknown. It involves the assumption of risk. It involves taking a view of the future.

Since pre-history, man has tried to predict the future. Ancient Greek soothsayers examined birds’ entrails and foretold on the basis of what they saw. Astrologers saw portents in the positions of celestial bodies. To this day, tarot readers use a deck of cards to gain insight into the future. Fortune tellers do it by examining the palm of your hand.

Insurance companies too need to form a view of the future. They need a secure basis for doing so. They take on risk in exchange for a premium. To survive, they need to be right more often than they are wrong. In order to achieve this, they use theories put forward not by soothsayers but by mathematicians. They use the laws of chance and probability.

The objective of this paper is to show how underwriters answer the question, what are the chances...? In grappling with this question, they are dealing with “real world” risks, such as the probability of automobile collision, fire, oil rig accident, earthquake, mine disaster, or hurricane. The techniques they use are backed by proven mathematical theorems. But this paper will not go too deeply into the mathematics. Suffice to say that the proof exists, both theoretically and in practice. Instead, we will review the practical application of the theories, where and how they are used in insurance.

The Concise Oxford Dictionary defines chance as “the way things happen of themselves; undesigned occurrence”, and probability as “the extent to which an event is likely to occur, measured by the ratio of the favourable cases to the whole number of cases possible”.

The simplest examples of the laws of probability are events which can end in two outcomes only – like tossing a coin, which can only land heads or tails. If the coin is evenly weighted, one toss has a 50% probability that the coin will land heads, and 50% tails. A more mathematical way of expressing this would be that the sum of a probability and its complement must always be 1. In other words, the probability of all possible outcomes adds to certainty.

So, if the probability that a fair coin will land “heads” on the first toss is 0.5, what is the probability that it will land “heads” both times on two tosses? The answer is 0.5 x 0.5 = 0.25. The more possible outcomes that there are, the more complicated the mathematics becomes.

An added complication is that different populations have different characteristics. We know, for example, that there is a greater probability of a hurricane in certain parts of the world than in others. Similarly, we know that people who live in certain countries are more likely to have a heart attack than others. Therefore, we need to sensibly segment the population from which we draw our data.

The larger the population of past outcomes from which data are drawn, the greater the certainty with which you can define the probability of the aggregate outcome. This axiom is much relied upon in insurance. It is known as “the law of large numbers”. Using the previous example of tossing a coin, every time you toss a coin, the probability of it landing heads is 0.5. But if you were to toss that coin 10 times, then progress to 100 times, then 1,000 times, the probability of your outcomes being 50% heads and 50% tails would approximate closer and closer to certainty. So the larger the population, the greater your ability to predict the aggregate result.

Another insurance application of the law of large numbers is in dealing with abnormally large claims. The reason why the insured takes out cover is to protect against unaffordable losses. The premiums of the many pay for the claims of the few.

One of the underwriter’s key decisions is setting the level of premium, known in the business as “rate making”. On that decision may depend whether the insurance company makes a profit or a loss. The basic purpose of rate making is to calculate an amount of premium per unit of exposure that will be adequate to pay the expected losses of the insureds in the risk pool. The formula commonly used is, premium = expected losses + acquisition costs + expenses + profit. In that formula, the biggest expense component is expected losses. In developing expected losses, the underwriter will use loss history data, and extrapolate, applying probabilities to various likely outcomes.

In using historical data to extrapolate future results, we should take into account outside variables which might influence the outcome. For example, the number of heart attacks in any given population may reduce as a result of improved diet, due to advertising; or as a result of advances in the medical field. We should factor variables which will have an effect on the data into our calculations.

Probabilities can usefully be applied in other fields in captive insurance management. When doing a risk assessment, for example, it may be informative to provide percentages of probability to each risk.

Another application for probability is in making a decision at what level to take out reinsurance. It may be cost effective to reinsure against catastrophe, while retaining the more predictable risks. A probability of occurrence figure would probably be helpful in making this decision.

Most captives invest their surplus capital. In constructing their investment portfolio, they are well advised to spread their investment risk. There is extensive mathematical theory behind minimizing the risk carried by an investment portfolio by spreading the risk, and that theory borrows heavily from probability theory.

As for future new developments in the use of probability theory by the captive insurance industry, one possibility is the application of Benford’s Law. This law states that in any set of naturally occurring numbers, the smaller the leading digit, the higher the chance of it occurring. This law is illustrated below.

Benford’s Law has found applications in the Internal Revenue Service as a screening device to spot check tax returns for indications of fraud. Plotting a histogram of each financial entry on a tax return and comparing its occurrence with the predicted frequency can raise a red flag. Financial auditors are doing the same thing.

In practice, much of the mathematical theory described in this article is used intuitively. An experienced underwriter may be using probability theory instinctively, based on his long experience. We live in an era when accumulated experience in many fields is being eliminated from companies and forced into retirement. In the insurance industry this has sometimes required inexperienced underwriters to be faced with rate making for a risk which is relatively new to them. One solution to this problem is to systematize the assessment of risk. Much of such systemisation would be based on probability theory.

In conclusion, we can say that owners, participants, and managers of captive insurance companies use probability theory regularly, whether consciously or unconsciously, in various applications. It gives a logical basis for many decisions that they have to make. Its use leads to better captive management, and makes it more likely that a captive will give its owner the desired result of a healthy profit.

Acknowlegements:

Westover, Kathryn A. “Captives and the Management of Risk” – International Risk Management Institute, Inc., Dallas, Texas.

www.IRMI-Online.com – “What are the Chances” and “The Laws of Probability”

Paul Brady – “The Island” © Hornall Brothers Music

The Concise Oxford Dictionary

Strong, Dr Robert A., University of Maine Business School “Benford’s Law: Leading Digit Probabilities, Accounting Fraud, and Tax Evasion.”