PROCEEDINGS May 15, 16, 17, 18, 2005 - Casualty Actuarial Society

100 WHY LARGER RISKS HAVE SMALLER INSURANCE CHARGES3. RISK-SIZE MODELSWe need to introduce some precision in our discussion to atleast guarantee that there is a well-defined notion of the insurancecharge for a particular risk size. To start, we initially ignoreparameter risk so that we can unambiguously identify the sizeof a risk with its expectation value. We then define a risk-sizemodel, M, as a collection of non-negative random variables eachhaving a finite non-negative mean. We index a random variablewithin such a model by its mean. We then use the risks of aparticular size in the model to define the cumulative distribution,limited expected value, and insurance charge at that size. We letM ¹ be the set of risks in M of size ¹ and we suppose there is ameasure § ¹ on M ¹ . We then define the cumulative distributionas a function of risk size via: F M (t j ¹)=E[F T (t) j T 2 M ¹ ]wherethe expectation is taken with respect to § ¹ . Similarly, we definelimited expected values and insurance charges as functions ofrisk size. We say M is well-defined if the measures give rise toa well-defined cumulative distribution for every M ¹ that is nonempty.We say a well-defined model is complete if there is acumulative distribution for the model at every size. Unless otherwisenoted, we henceforth assume all models are well-definedand complete. We define M to be size continuous at t>0ifF M (t j ¹) is a continuous function of ¹ and nth order size differentiableat t>0ifF M (t j ¹) hasanth order partial derivativewith respect to ¹, for¹>0. Note that M could be size continuousand differentiable even if all the random variables in M arediscrete.In the simplest case, each M ¹ consists of a single randomvariable that we denote as T ¹ , and the measure, § ¹ , assigns a masspoint of 100% to this random variable. We say this is a uniquesize model and we use F T¹(t), the cumulative distribution functionfor the unique risk of size, ¹, todefineF M (t j ¹), the cumulativedistribution function at t for the model at size ¹. Similarly we usethe limited expected value and charge function of T ¹ .todefine