Commercial Insurance in One Page ================================ A five-minute primer on the handful of commercial-insurance terms you will meet throughout this guide, the example notebooks, and the HJB optimization experiment. It is written for readers who are already comfortable with the *ergodic* idea -- time-average growth versus ensemble average -- but who have never had to read a commercial insurance program. If a chart or notebook says "buy down the SIR on a \$200M tower," this page is all the vocabulary you need to follow it. You do not need any of this to understand *why* insurance helps (see :doc:`executive_summary`); you need it to follow *how* a real program is built, priced, and optimized. The one picture that unlocks everything: the tower -------------------------------------------------- A commercial insurance program is rarely a single policy. It is a **stack of layers**, drawn as a vertical *tower*, each layer covering a different band of loss. The company keeps the very bottom band itself; insurers cover the bands above, up to some maximum. Above the top, the company is exposed again. Here is the exact tower used in the notebook 07 experiment, for a \$5M manufacturer (an illustrative structure -- the dollar figures are specific to that example): .. code-block:: text $200M +---------------------------+ <- top of program | Layer 4: Catastrophic | | $150M xs $50M | rare, near-existential events $50M +---------------------------+ | Layer 3: 2nd Excess | | $25M xs $25M | infrequent, severe losses $25M +---------------------------+ | Layer 2: 1st Excess | | $20M xs $5M | large losses $5M +---------------------------+ | Layer 1: Primary | | $4.75M xs $250K | the "working layer" (frequent) $250K +---------------------------+ | Self-Insured Retention | the company pays this band itself $0 +---------------------------+ Read each layer as **"limit xs attachment"**: "\$20M xs \$5M" means a limit of \$20M sitting *excess of* (on top of) a \$5M attachment point -- so it covers the band from \$5M to \$25M. How a single loss flows through the tower ----------------------------------------- Follow one loss up the stack. Each band pays only its own slice: * **Loss below \$250K** -- the company pays in full (this is the retention). * **\$250K to \$5M** -- Layer 1 (Primary) responds, after the company's first \$250K. * **\$5M to \$25M** -- Layer 2 (1st Excess) picks up the next band. * **\$25M to \$50M** -- Layer 3 (2nd Excess). * **\$50M to \$200M** -- Layer 4 (Catastrophic). * **Above \$200M** -- uninsured; the company is fully exposed again. So a \$30M loss is split: \$250K (company) + \$4.75M (Layer 1) + \$20M (Layer 2) + \$5M (Layer 3) = \$30M, with Layer 4 left untouched. The vocabulary, defined against that picture -------------------------------------------- **Self-Insured Retention (SIR) / Retention / Deductible** The amount of each loss the company keeps before any insurance pays -- the bottom band of the tower. In this framework the three terms are used interchangeably, and the SIR is **the one knob the optimizer turns**: raise it and you keep more risk but pay less premium; lower it and insurance does more work for a higher price. (In real contracts a true SIR and a deductible differ in who administers the claim and posts collateral; for our purposes they are the same lever.) **Layer** One horizontal band of coverage, defined by an attachment point and a limit. **Attachment point** The loss level at which a layer *starts* paying. Layer 2 above attaches at \$5M. **Limit** The most a layer -- or the whole program -- will pay. Layer 2's limit is \$20M, so it covers the \$5M-\$25M band and no more. **Primary, Excess, Catastrophic (Cat)** The **primary** layer sits just above the retention and absorbs the frequent, smaller losses (the "working layer"). **Excess** layers stack above it for larger, rarer losses. The **cat** layer at the top is for rare, near-existential events. **Tower / Program** The full stack from the retention up to the top limit (\$200M here). A loss above the top is retained by the company. **Premium** The annual price paid for a layer, or for the whole tower. **Rate-on-line (ROL)** Premium divided by a layer's limit -- the price per dollar of coverage. It **falls steeply as you climb the tower**: the primary layer is hit most years (high ROL), while the cat layer almost never attaches (low ROL). **Loss ratio** Expected losses divided by premium. A 65% loss ratio means 65 cents of every premium dollar is expected to return as losses; the other 35 cents is the insurer's **loading** for expenses, capital, and profit. Lower layers carry higher loss ratios; tail layers are loaded more heavily for their volatility. **Frequency-severity model** How losses are generated: a **frequency** distribution for *how many* events occur per year (here, Poisson) and a **severity** distribution for *how big* each one is (lognormal for ordinary losses, heavy-tailed Pareto for catastrophes). You already know these as ordinary probability; insurance simply separates "how often" from "how bad." Why this matters for ergodic optimization ----------------------------------------- Everything above is structural plumbing. The reason the framework cares is that the **retention is a growth decision, not merely a cost decision**. A lower SIR drags on the survivors' growth -- you pay more premium every year -- but it sharply cuts the chance of a loss large enough to end the company. Because a single firm lives *one* path through time, avoiding ruin can raise its time-average growth rate even when the premium exceeds expected losses -- the central result this documentation develops. The HJB experiment in notebook 07 simply searches for the SIR, as a function of the firm's size and leverage, that maximizes that time-average growth. .. note:: **Want the formal definitions?** The :doc:`glossary` lists every term alphabetically. For the live example this page is drawn from, see Part 3 of the `notebook 07 HJB experiment `__.