Insurer Probabilities of Profits by Portfolio Size

Insurers use the 85% of their premiums not allocated to operating expenses in Formula 1 to pay policyholders' health expenses, converting all unused portions to profits. Table 1 Row 3, shows all insurers have probability, $\Phi_N(0.7500)$ = 0.5000, of profits of at least 10%, at PLREs at, or below, PLR (0.7500), because E[PLRE] = PLR for all insurers. Capitation advocates may not have gone any further than this because this is obviously the only PLRE value for which insurers' profit probabilities are identical. Small insurers have more probability in the tails of their distributions which is why they tend to have volatile operating results. Large insurers tend to have most of their probability close to the PLR, so their operating results tend to vary very little.

Table 1 Row 4, shows insurers' probabilities of profits of at least 5%, ( $\Phi_N(0.8000)$), at PLREs below 0.8000. $NHI$ earns such profits with probability 1.0000, $B$ with probability 0.9992, and $PI$ with probability 0.8413. $D$ and $E$ have much lower probabilities of earning such profits, 0.6241 and 0.5398, respectively. However, $\Phi_{NHI}(0.8000)$) = 1.0000 is very misleading. $NHI$'s probability, $\Phi_{NHI}$(PLR + 3 * $\sigma_{e_{NHI}}$) = $\Phi_{NHI}(0.758547)$ = 0.9987. $NHI$ almost always earns profits greater than 9.15%, and since $\Phi_{B}(0.79743)$ = 0.9987, $B$ almost always earns profits greater than 5.25%!

Table 1 Row 5, shows insurers' probabilities of profits greater than 0% (Break Even), $\Phi_N(0.8500)$, at PLREs below 0.8500. $NHI$ and $B$ have probability 1.0000, $PI$'s probability is 0.9772, but $D$ and $E$ have much lower ``break-even`` probabilities, 0.7365 and 0.5793, respectively.