Chapter 10 — Key Takeaways

A one-page field card for The Mathematics of Risk and Price. All figures are illustrative.

The core claims

• Expected loss = frequency × severity, and each is a distribution, not a number. Keep them apart when you read a loss run — a single losses-÷-years figure hides which account you are pricing.
• Frequency is low-count and lumpy (often Poisson): zero claims in a year tells you almost nothing. Severity is right-skewed with a long tail: the mean claim sits well above the typical (median) claim, pulled up by rare large losses.
• The loss ratio tells the truth only as incurred losses ÷ earned premium. Paid-over-written flatters a young, growing, long-tail book — the losses haven't arrived or developed and the premium hasn't earned.
• The pure premium (loss cost) is the expected loss per exposure — the honest, expense-and-profit-free core of every rate. A good exposure base varies with loss and resists manipulation (value, payroll, vehicles).
• Trend and development restate old, immature losses to the period you're actually pricing. Development grows immature claims toward ultimate (up); trend moves them to future cost levels (up). Skipping them is how soft markets manufacture future losses.
• Credibility ($Z$, from 0 to 1) is how much you trust a risk's own experience versus its class. A small, skewed sample earns only a small $Z$. The square-root rule: $Z = \sqrt{n/N}$ — quadrupling the data only doubles the credibility.
• Credibility weighting blends the two: $\text{Estimate} = Z \times \text{own} + (1-Z) \times \text{class}$. It guards against both over-reacting to a risk's own bad experience and ignoring its own experience entirely.
• Credibility math is blind to causation. It treats experience as random noise around a fixed risk. When the qualitative story says the risk changed (a live, recurring hazard), the underwriter overrides the blend — upward, with documented reasons. That is judgment.

The key formulas / rules of thumb

  expected loss            = frequency × severity
  loss ratio (honest)      = incurred losses ÷ earned premium
  pure premium (loss cost) = total losses ÷ exposures   ( = frequency × severity )
  permissible loss ratio   = 1 − (expense load + profit/contingency load)
  charged rate (approx.)   = pure premium ÷ permissible loss ratio
  trended/developed loss   = raw loss × development factor × (1 + trend)^years
  classical credibility    = Z = sqrt(n / N)        (N = full-credibility claim standard)
  Bühlmann credibility     = Z = n / (n + K),  K = EPV / VHM
  credibility-weighted     = Z × own + (1 − Z) × class

Rule of thumb: two claims is a single-digit $Z$. Run the credibility number before you let a couple of losses move your price.

Key terms

pure premium · loss cost · credibility · full credibility · partial credibility · credibility weighting · frequency distribution · severity distribution · expected loss · trend and development

What you could defend to your manager

"I am not pricing Harbor Steel off its raw two-fire history. Two fires is low credibility — the square-root rule puts the account's own fire experience in single digits, so the class loss cost deserves most of the weight, and the credibility-weighted expected number lands far below the naive losses-÷-years figure. That said, the 2023 fire was hot-work-related — a live, recurring hazard, not random noise — so I'm overriding the credibility blend upward on that exposure and handling it through subjectivities and terms rather than the base rate. The math tells me how much the data has earned the right to say; my read of the loss run handles the part the math is blind to."