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 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):

$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 Glossary lists every term alphabetically. For the live example this page is drawn from, see Part 3 of the notebook 07 HJB experiment.