Efficient Insurer Portfolio Selections Are Random Samples

We know that insurer PLREs are uncertain because policyholders' health experiences vary. Very high PLREs occur when one, or more, policyholders incur high Claims Costs. Very low PLREs occur when very few policyholders incur high Claims Costs. But ``perfect'' years are years in which insurers' PLREs are exactly equal to the Population Loss Ratio. A perfectly efficient insurer does not have Claims Costs of $0.00. Perfectly efficient insurers issue infinitely many policies, and because $\sigma_{e_{\infty}}$ = 0.0000, their Claims Costs equal their Expected Claims Costs. Inefficient insurers have high, or low, Claims Costs but their ``inefficiency'' is due to their high Claims Costs variation (i.e. large standard errors).

Insurance premiums vary with policyholder risk characteristics and in efficient insurance markets, no insurer can systematically select lower risk policyholders. If this happens high risk policyholders may go uninsured, the most selective insurers get excessive premiums, and the least selective insurers, with too many high risk policyholders, get inadequate premiums. In efficient insurance markets, insurers randomly select policyholders.

Efficient insurance markets are stochastic processes subject to the Central Limit Theorem (CLT) (Hogg and Craig, 1978). PLREs are ``averages'' calculated from large, randomly selected groups of policyholders, and PLRE Cumulative Distribution Functions are normally distributed. I can analyze how portfolio size affects insurers' PLRE variability and operating results by specifying the Population Loss Ratio and either the standard error for a single, Paradigm Insurer ($PI$), or the standard deviation for an individual policyholder. Once these are specified, the CLT describes other insurer's standard errors and PLRE Distribution Functions, which I use to calculate probabilities of PLREs and operating results for insurers of any size.

Two insurers, $\textbf{\textit{M}}$ and $\textbf{\textit{N}}$, randomly selecting $M$ and $N$ ($M » N$) policyholders from the same population, with individual policyholder PLRE standard deviation, $\sigma$, and Population Loss Ratio, $\mu$, draw PLREs from very different, normally distributed Cumulative PLRE Distribution Functions: $\Phi_{M}$($\mu$, $\frac{\sigma}{\sqrt{M}}$) and $\Phi_{M}$($\mu$, $\frac{\sigma}{\sqrt{N}}$), where $\frac{\sigma}{\sqrt{M}}$ $«$ $\frac{\sigma}{\sqrt{N}}$ and each insurer's standard error determines all other insurers' standard errors since $\sigma_{e_{M}}$ = $\sigma$ * $\frac{\sqrt{N}}{\sqrt{M}}$.

I will specify the Population Loss Ratio (PLR) and a reasonable, market appropriate, assumption about Population Loss Ratio Estimate (PLRE) variation for a single, ``reasonably efficient'' Paradigm Insurer ($PI$), then analyze, and compare, the impact of portfolio size on operating results for four other insurers. The ``Market Premium'' for this reasonably efficient insurer is an adequate, but not excessive, expense, risk, and profit loaded premium such that if all policyholders pay the ``Market Premium,'' the expected value of the total industry PLRE equals the PLR, and the market will continue to operate. Reasonable efficiency means that when the Paradigm Insurer receives the market premium, it can continue to operate with minimal risk of insolvency and acceptable levels of profits.