the portfolio percentile premium
Setting the premium so the insurer just breaks even on average (the equivalence principle) is fair but dangerous: 'on average' means that about half the time the actual outcome will be worse than expected, and a string of bad luck could sink the company. A more cautious idea is to ask: how much should we charge so that, looking at the whole book of business, we are 95% sure the premiums will be enough to cover the claims? That target charge is the portfolio percentile premium.
Precisely, consider the total loss of a whole portfolio of n similar, independent policies — the sum of the individual loss-at-issue variables, call it S. With the net premium, E[S] = 0. The portfolio percentile premium raises the premium until the probability that S stays below zero (i.e., the portfolio does not lose money) reaches a chosen confidence level, say 95%. Using the central limit theorem, S is approximately normal, so you set the premium so that mean-plus-1.645-standard-deviations of S equals zero. Crucially, because the policies are independent, the standard deviation of S grows like the square root of n while the mean grows like n, so the extra loading per policy SHRINKS as the portfolio gets bigger — the law of large numbers at work.
This principle makes concrete why a bigger pool is safer and how much security loading a given confidence level costs. It underlies risk loadings, capital requirements, and the intuition behind solvency margins. An honest caveat: it leans on independence and a normal approximation. Real portfolios have correlated risks — a pandemic, a market crash, a systematic mortality misestimate — that hit many policies at once, so the diversification benefit is smaller than the clean square-root rule suggests, and prudent insurers add catastrophe and parameter-risk margins on top.
For 10,000 independent policies, the portfolio's standard deviation is only 100 times one policy's (√10,000 = 100), but its mean would scale by 10,000 — so a tiny per-policy loading buys 95% confidence. Shrink the pool to 100 policies and you need a far heavier loading for the same safety.
Charge enough that the whole portfolio is, say, 95% likely to come out ahead — loading per policy shrinks with size.
The square-root diversification only works for INDEPENDENT risks. Correlated shocks (pandemics, market crashes, a wrong mortality table) hit the whole book at once and do not diversify away.