Non-Life Insurance Mathematics

Actuaries are the people who deal with all kinds of mathematical and statistical problems in insurance—that’s why we speak of actuarial problems. With the more recent application of actuarial methods also to Property and Liability insurance, it has become customary to distinguish between Life and Non-Life actuarial sciences. Although such a distinction does not always make sense — in Health insurance, for instance, the two domains overlap widely — the present book deals exclusively with so-called Non-Life problems and their possible solutions.

In this chapter those mathematical tools are put together which are needed for the solution of the problems described above. Also, most of the basic notations used throughout the book will be introduced here. In the following, an attempt has been made to organise the main tools into six different domains which may be viewed as selected parts from general probability theory. Prerequisites are college mathematics only, more precisely elements of calculus and probability.

The premium is the price for the good “insurance” (or “reinsurance”) sold by the insurance industry. So, as with any other industry, the right pricing—which is here called “rating” — is vital since too low a price level results in a loss, while with too high rates a company or a whole sector of the insurance industry can price itself out of the market. On top of this, premiums and tariffs are often a political subject, particularly so of course with lines of business such as Social Security, Motor Liability and Health insurance. And finally also the insurance supervisory authorities have a peculiar interest in premiums and rate levels since, if they approve too low prices, they would share the guilt if a company went bankrupt and, if they support inappropriately high rates, they could be criticised for helping the industry become rich at the cost of the man in the street.

What is reinsurance and why reinsure? Reinsurance is, broadly speaking, the insurance of insurance companies. If an individual risk is too big for an insurance company, or if the loss potential of its entire portfolio is too heavy — then the company either decides to or is forced to buy reinsurance protection. Often the reinsurance company does the same, i.e. it retrocedes part of a risk or parts of its portfolio to a third company. By passing on parts of risks, large risks particularly are finally split up into a number of portions placed with many different risk carriers. The same happens in real life: whenever there is a catastrophe such as an earthquake, a windstorm or an airline crash, there is usually a large number of insurance and reinsurance companies involved, each of them paying their share of the total insured loss according to the specific conditions of their policies and/or reinsurance in force at the time of the occurrence of the said catastrophe.

The problem of how to fix retention is quite complex on the one hand because a number of different criteria like solvency, capacity, financing, services, reciprocity and levelling out the net result have an impact on the retention and, on the other hand, because in practice, as a rule, we are not confronted with the question of how to fix one single retention but rather how to design in a somehow optimal way an entire insurance or reinsurance programme.

It is generally known that we look into the past in order to guess what the future will bring. The same is true in insurance. As we have, for example, to fix the premium in advance for a risk or a collective of risks, we naturally look at claims that occurred in the past on similar risks. We compile statistics to get an idea of the future claims potential. It is crucial that this picture be realistic because, if our judgement of the future claims load is too low, we will very probably suffer a loss. If, on the contrary, we are too pessimistic, we may lose business, e.g. where our competitors are cheaper.

In insurance and reinsurance there are many different types of reserves, such as premium reserves, claims reserves, catastrophe reserves, contingency reserves, currency fluctuation reserves, IBNR-reserves, additional case reserves and — as can be found in almost every balance sheet — of course all kinds of pretty well undefined “special” reserves.

The solutions have practically all been given in the preceding chapters so that we can confine ourselves here to picking up a few remaining odds and ends and closing with some summarising final remarks.