Context: risk theory, actuarial science, insurance

The adjustment coefficient is used extensively in risk theory as a rudimentary measure of risk in a collective of insurance risks. The smaller the adjustment coefficient, the more likely that an insurance company would not be able to meet claim payments, i.e. become ruined.

The adjustment coefficient *r* is the positive root of

*λM*_{X}(*r*) = *λ* + *cr*,

where

*λ* is the parameter in the claims arrival

Poisson process (an indicator on how

*often* claims arrive),

*c* is the

premium rate charged, and

*M*_{X}(

*t*) is the

moment generating function of the claim amount, assuming that the claim amounts are identically distributed.

Finding out the adjustment coefficient is important for actuaries as it helps them to decide how much premiums to charge, and whether to take out reinsurance, and in what form.

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