Appendix A: Key Formulas and Ratios
This is the working reference you keep open on the second monitor: every load-bearing formula and ratio in the book, in one place, in the order you actually meet them on the desk — from the loss ratio that judges a book to the return period that sizes a catastrophe program. For each one you get three things and only three: the formula, a one-line plain-language meaning (what the number is telling you), and a short worked example that shows it producing an answer. Every number in every example is a constructed teaching example — a round figure chosen to make the arithmetic legible, not a real insurer's experience. Where an example reuses figures from the chapter that owns it, it is so noted, so the worked numbers stay consistent across the book.
A word on how to read this appendix, because it is the same word the whole book has pressed: a formula is the answer to an underwriting question, never a procedure handed down from nowhere. The loss ratio answers "are we making money on this business?" The pure premium answers "what is this risk expected to cost before we add anything?" The X-mod answers "how much should this employer's own history move its price?" Memorize the question first; the formula is just the question written in symbols. And remember the recurring discipline: most of these numbers are estimates with error bars, not facts. The combined ratio is backward-looking and the loss ratio is only as good as the reserves inside it; the PML is a model output, not a measurement; credibility tells you how little a small sample is worth. Use the number, then read behind it.
Inline math uses single-dollar delimiters; currency in the prose is escaped as \$.
1. The ratios that judge the result (Chapter 3)
These are the scoreboard. They are computed after the business is written, and they tell you whether the underwriting was any good. The combined ratio is the one the whole book calls "the truth."
Loss ratio
$$\text{Loss ratio} = \frac{\text{incurred losses} + \text{loss adjustment expense}}{\text{earned premium}}$$
Meaning. The share of every premium dollar that went to paying claims (and the cost of adjusting them). It is the part of the result the underwriter most directly controls, and the part most contaminated by reserve estimates — a loss ratio is only as honest as the reserves inside "incurred losses."
Worked example (constructed teaching example). A book earns \$10,000,000 of premium and incurs \$6,200,000 of losses and loss adjustment expense.
LOSS RATIO [constructed teaching example]
incurred losses + LAE ............ $6,200,000
earned premium ................... $10,000,000
loss ratio = 6,200,000 / 10,000,000 = 0.62 = 62%
Read it: 62 cents of every premium dollar paid for losses; 38 cents is left for expenses and profit.
Expense ratio
$$\text{Expense ratio} = \frac{\text{underwriting expenses (acquisition} + \text{general)}}{\text{premium}}$$
Meaning. The share of premium consumed by the cost of running the business — chiefly commission, plus underwriting, issuance, premium taxes, and overhead. (Statutory presentation can divide by written premium and loss ratio by earned premium; for the worked examples here we hold premium constant so the ratios add.)
Worked example (constructed teaching example). On the same \$10,000,000 of premium, the carrier spends \$3,000,000 on commissions, operations, taxes, and overhead.
EXPENSE RATIO [constructed teaching example]
underwriting expenses ............ $3,000,000
premium .......................... $10,000,000
expense ratio = 3,000,000 / 10,000,000 = 0.30 = 30%
Combined ratio
$$\text{Combined ratio} = \text{loss ratio} + \text{expense ratio}$$
Meaning. The share of every premium dollar paid out in losses and expenses. Below 100% is an underwriting profit; above 100% is an underwriting loss — before any investment income. A combined ratio of 104% means the insurer paid out \$1.04 in losses and expenses for every \$1.00 of premium.
Worked example (constructed teaching example, the numbers above).
COMBINED RATIO [constructed teaching example]
loss ratio ....... 62%
expense ratio .... 30%
combined ratio = 62% + 30% = 92%
Read it: an 8-point underwriting profit — the book keeps about 8 cents of every premium dollar
on underwriting alone, before a dollar of investment income. A 92% is a healthy commercial result.
Operating ratio
$$\text{Operating ratio} = \text{combined ratio} - \text{net investment income ratio}$$
Meaning. The combined ratio after crediting the investment income earned on the float (the premium and reserves held before claims are paid). It is why a line can run a combined ratio slightly above 100% and still be profitable overall — the investment income closes the gap. It does not rescue a badly underpriced book; it narrows an honest one.
Worked example (constructed teaching example). A long-tail book runs a 103% combined ratio but earns investment income equal to 6% of premium on its float.
OPERATING RATIO [constructed teaching example]
combined ratio ............... 103% (a 3-point underwriting LOSS)
− investment income ratio .... 6%
operating ratio = 103% − 6% = 97% (a 3-point operating PROFIT)
Read it: the underwriting lost money; the investment income on the float more than covered it.
The trap (Ch.3): relying on this turns underwriters into bond traders and hides a pricing problem.
2. The structure of expected loss (Chapters 6 and 10)
Before there is a price, there is an expected loss. The book's single most important structural idea is that expected loss has two dimensions — how often, and how big.
Pure premium = frequency × severity
$$\text{Pure premium} = \text{expected frequency} \times \text{expected severity} = \frac{\text{total losses}}{\text{exposure units}}$$
Meaning. The expected loss per unit of exposure — the irreducible cost of the risk, before a single dollar of expense or profit. It is the floor below which no rate can go without the insurer paying claims out of its own capital. The rating bureaus (ISO/Verisk, NCCI) publish this same quantity by class and call it the loss cost.
Worked example (constructed teaching example). A class of 4,000 similar buildings produces, on average, 0.05 fire losses per building per year (a 1-in-20 chance), and the average fire loss is \$80,000.
PURE PREMIUM = FREQUENCY × SEVERITY [constructed teaching example]
expected frequency ... 0.05 losses per building-year
expected severity .... $80,000 per loss
pure premium = 0.05 × $80,000 = $4,000 per building per year
Equivalently, per $1,000 of a $1,000,000 building value: $4,000 / 1,000 = $4.00 per $1,000.
That $4.00 is the pure-premium block in the build-up below.
A note on the two dimensions (Ch.6). Frequency ("how many") is the more stable, more predictable number and the one the insured most controls through loss prevention. Severity ("how big") is fat-tailed and driven by rare large losses; its maximum, not its average, is what decides catastrophe survival. This is why so many rating plans — the X-mod above all — are built to respond to frequency.
3. Building the rate (Chapter 11)
The premium is built, from the bottom up, out of three blocks, then adjusted for the individual risk. The figures below are the canonical build-up used throughout Chapter 11.
The premium build-up
$$\text{Indicated rate} = \text{pure premium} + \text{expense load} + \text{profit and contingencies load}$$
Meaning. Stack the expected loss, the cost of doing business, and the margin (plus a cushion for being wrong), and you have the indicated rate — the price the math says this class should pay, before any adjustment for the specific risk. A rate missing any one block is not a rate; it is a subsidy.
Worked example (constructed teaching example, Ch.11 figures).
PREMIUM BUILD-UP ($ per $1,000 of building value) [constructed teaching example]
pure premium (expected loss) ████████████████ $4.00 ← frequency × severity (§2 above)
+ expense load (~30% of premium) ███████ $1.85 ← commissions, ops, taxes, overhead
+ profit & contingencies (~5%) ██ $0.40 ← margin + cushion for being wrong
─────────────────────────────────────────────────
= indicated manual rate $6.25 (before experience/schedule modification)
On a $20,000,000 building: $6.25 × 20,000 = $125,000 indicated building premium (illustrative).
The permissible loss ratio
$$\text{Permissible loss ratio} = 1 - (\text{expense ratio} + \text{profit and contingencies load})$$
Meaning. The share of each premium dollar that is allowed to go to losses after expenses and profit are removed. The rate is set so the pure premium fills exactly this and no more. It is why the same loss ratio means different things in different lines: a 75% loss ratio is fine where expenses are low and a disaster where commissions are high.
Worked example (constructed teaching example). Expenses run 30% and target profit and contingencies 5%.
PERMISSIBLE LOSS RATIO [constructed teaching example]
expenses + profit/contingencies = 30% + 5% = 35%
permissible loss ratio = 1 − 0.35 = 0.65 = 65%
Read it: the rate is built so expected losses are 65% of premium. If actual losses run 72%,
the 5-point profit load is gone and then some — the account is underwriting at a loss.
The loss-cost multiplier (LCM)
$$\text{Rate} = \text{loss cost} \times \text{loss-cost multiplier (LCM)}$$
Meaning. Bureaus file the loss cost (the expected-loss component only). Each carrier turns it into a chargeable rate by multiplying by its own LCM, which grosses the loss cost up for that carrier's expenses and profit target. It is why two insurers writing the same class from the same loss cost charge different rates — different expense structures, different LCMs. The classic trap is quoting the loss cost as the rate, selling coverage with zero expense load and zero profit.
Worked example (constructed teaching example, Ch.11 figures).
LOSS COST → RATE [constructed teaching example]
ISO/NCCI loss cost (expected loss only) ... $4.00 per $1,000 ← filed by the advisory org
× insurer's loss-cost multiplier .......... × 1.56 ← THIS carrier's expense & profit
= insurer's manual rate ................... $6.25 per $1,000 ← what the carrier files and charges
The $1.56 implies a permissible loss ratio of 1 / 1.56 ≈ 0.64 — the same ~65% as the build-up above.
Rating factors (relativities)
$$\text{Risk rate} = \text{base rate} \times f_1 \times f_2 \times \cdots \times f_k$$
Meaning. Each rating factor (relativity) is a multiplier that turns one risk characteristic into price: 1.00 is baseline, above 1.00 a debit (more expected loss), below 1.00 a credit (less). Read a string of them as a sentence about expected loss, not as arithmetic.
Worked example (constructed teaching example, Ch.11 figures).
RATING-FACTOR BUILD-UP — a light-manufacturing building [constructed teaching example]
base rate (standard construction, protection) $3.50 per $1,000
× construction (joisted masonry, not fire-resistive) × 1.15
× protection class (Class 4, not Class 1–2) × 1.10
× occupancy hazard (metal fabrication, hot work) × 1.30
× sprinkler credit (wet-pipe system present) × 0.90
──────────────────────────────────────────────────────
= building rate before experience/schedule mod $5.18 per $1,000 (3.50×1.15×1.10×1.30×0.90)
4. Adjusting the class rate to the individual risk (Chapters 11 and 22)
The class rate prices the class. The next two plans pull it toward this risk — one by formula (experience rating), one by judgment (schedule rating).
Rate adequacy and the rate indication
$$\text{Indicated rate change} = \frac{\text{trended, developed loss ratio}}{\text{permissible loss ratio}} - 1$$
Meaning. Rate adequacy is the discipline that the price must cover expected losses, expenses, and a reasonable margin for the risk accepted. The rate indication is the actuary's verdict on whether the current rate level clears that bar: compare the experience loss ratio (properly trended and developed) to the permissible loss ratio. Above 1.00, the rate is inadequate and needs to rise; below 1.00, there is room.
Worked example (constructed teaching example). A book's trended, developed loss ratio is 72%; the permissible loss ratio is 65%.
RATE INDICATION [constructed teaching example]
experience (trended, developed) loss ratio .. 72%
permissible loss ratio ...................... 65%
indicated change = 72% / 65% − 1 = 1.108 − 1 = +10.8%
Read it: the rate is running about 11% short of adequacy. Holding the line on that increase —
against a broker pushing back in a soft market — is the hardest discipline in insurance (Ch.11).
Experience rating (general form)
$$\text{Modified premium} = \text{manual premium} \times \text{experience modification}$$
Meaning. Experience rating adjusts the manual premium up or down by the risk's own loss experience relative to what its class would predict — weighted by how credible that experience is (Chapter 10). The workers'-comp X-mod is its canonical, filed form.
The workers'-comp experience modification (X-mod)
$$\text{X-mod} \approx \text{credibility-weighted ratio of } \frac{\text{actual losses}}{\text{expected losses}}, \qquad \text{Modified premium} = \text{manual premium} \times \text{X-mod}$$
Meaning. A single filed multiplier that turns an employer's own multi-year loss history into its price. 1.00 is the class average; below 1.00 is a credit mod (a discount for better-than-average experience); above 1.00 is a debit mod (a surcharge). The published formula (NCCI or a state bureau) uses roughly the last three years, splits every claim into a heavily weighted primary slice and a lightly weighted excess slice, and so responds far more to claim frequency than to severity — many small claims move the mod more than one large claim of the same total dollars. You do not invent the mod; you read it, and you read behind it.
Worked example (constructed teaching example). Two welding shops carry the manual premium below.
THE X-MOD APPLIED [constructed teaching example]
manual premium (from class rates × payroll) ... $200,000
Shop A — credit mod 0.85 → modified = 200,000 × 0.85 = $170,000 (a 15% discount; better-than-class)
Shop B — debit mod 1.25 → modified = 200,000 × 1.25 = $250,000 (a 25% surcharge; worse-than-class)
Read behind it (Ch.22): if Shop B's 1.25 comes from THREE small claims, it is a frequency pattern
(a workplace that keeps hurting people) — a worse forward signal than one large fluke claim.
Schedule rating (credits and debits)
$$\text{Scheduled premium} = \text{manual premium} \times \left(1 + \sum \text{debits} - \sum \text{credits}\right)$$
Meaning. Documented credits (reductions) and debits (increases) for risk characteristics the class rate and experience rating cannot see — management quality, premises and protection, equipment, loss-control programs. Most plans cap the net swing (commonly around ±25% — treat the exact cap as filing-specific). Schedule rating is prospective and judgmental; experience rating is retrospective and formulaic. A risk can carry a debit X-mod for past losses and a schedule credit for the new controls that change the future — no contradiction.
Worked example (constructed teaching example). A property account earns a 10% debit for an end-of-service-life roof and a 5% debit for poor housekeeping, partly offset by a 5% credit for a strong, verified safety culture.
SCHEDULE RATING [constructed teaching example]
manual premium ........................ $125,000
+ roof debit (near-term wind/water) ... +10%
+ housekeeping debit .................. +5%
− safety-culture credit (verified) .... −5%
net schedule modification = +10% + 5% − 5% = +10% (a net DEBIT)
scheduled premium = $125,000 × 1.10 = $137,500
Discipline (Ch.11): a credit you cannot document is a soft credit — and soft credits are how a book
gets underpriced one defensible-looking concession at a time. Verify the control before you credit it.
5. Structuring the deal: coinsurance and insurance-to-value (Chapters 12, 15, 19)
These belong to the terms, not the price — the clauses that decide what a partial loss actually pays.
The coinsurance penalty
$$\text{Loss payment} = \text{loss} \times \frac{\text{insurance carried}}{\text{insurance required}} - \text{deductible}$$
where insurance required = coinsurance percentage × the property's full value at the time of loss, and the fraction is capped at 1.
Meaning. A property condition that punishes under-insurance. If the insured carries less than the required percentage (commonly 80/90/100%) of full value, the insurer pays only a proportional share of every loss — even a partial loss far below the limit. It exists to force insurance-to-value and defeat the adverse-selection move of buying only enough limit for the partial losses one expects.
Worked example (constructed teaching example, Ch.12 figures). An 80% coinsurance clause; a \$10,000,000 building; the insured carries \$6,000,000; a \$2,000,000 fire; a \$50,000 deductible.
THE COINSURANCE PENALTY — an 80% coinsurance clause [constructed teaching example]
full value at time of loss ......... $10,000,000
required (80% of value) ............ $8,000,000 ← insurance REQUIRED
carried ............................ $6,000,000 ← insurance CARRIED
partial loss ....................... $2,000,000
deductible ......................... $50,000
penalty ratio = carried / required = 6,000,000 / 8,000,000 = 0.75
loss payment = $2,000,000 × 0.75 − $50,000 = $1,500,000 − $50,000 = $1,450,000
The loss ($2M) was far below the carried limit ($6M) — yet the insured collects only $1,450,000.
The missing $550,000 is the coinsurance penalty. The limit was never the problem; the VALUE was.
Insurance-to-value (ITV)
$$\text{Insurance-to-value} = \frac{\text{limit carried}}{\text{full replacement cost of the property}}$$
Meaning. The principle (and the policy requirement) that the limit equal the full cost to rebuild — not purchase price or market value, which include land and location. The coinsurance and replacement-cost conditions enforce it; chronic under-ITV, worsened by construction-cost inflation and post-catastrophe demand surge, is the property line's quiet structural weakness — and an underwriting problem (you collected too little premium all year) as much as a claims one.
Worked example (constructed teaching example). A home costs \$500,000 to rebuild; the owner insures it for \$400,000 because that is roughly the market price.
INSURANCE-TO-VALUE [constructed teaching example]
limit carried ............... $400,000
full replacement cost ....... $500,000
ITV = 400,000 / 500,000 = 0.80 = 80% insured to value
Under an 80% coinsurance/replacement-cost condition this is just barely compliant; a year of
rebuild-cost inflation pushes replacement cost to $540,000 and the same $400,000 limit silently
falls to 74% ITV — a penalty waiting at the next partial loss.
6. How much to trust the data: credibility (Chapter 10)
Credibility theory is the humility the math demands about small samples — the antidote to over-reacting to a risk's own bad year and to ignoring a genuine signal.
The credibility-weighted estimate
$$\text{Estimate} = Z \times (\text{risk's own experience}) + (1 - Z) \times (\text{class expectation})$$
Meaning. Blend the risk's own indicated experience with the class benchmark, weighting the risk's own data by its credibility $Z$ (a number between 0 and 1). The formula listens to the account's own experience exactly as much as it deserves — no more, no less.
Worked example (constructed teaching example, Ch.10 figures). An account's own trended, developed loss ratio is 95%; its class runs 62%; you assess $Z = 0.30$.
CREDIBILITY-WEIGHTED LOSS RATIO [constructed teaching example]
own loss ratio (trended, developed) .. 95% (looks like a problem)
class loss ratio ..................... 62% (the benchmark)
credibility .......................... Z = 0.30
weighted = Z × own + (1 − Z) × class
= 0.30 × 95% + 0.70 × 62%
= 28.5% + 43.4% = 71.9% ≈ 72%
Read it: the bad history pulls the estimate UP from the 62% class — but only partway, because at
30% credibility the account's own small-sample experience earns only 30% of the vote.
The square-root partial-credibility rule
$$Z = \sqrt{\frac{n}{N}}$$
where $n$ is the number of claims observed and $N$ is the full-credibility standard (the claim count at which $Z = 1$).
Meaning. Credibility does not grow linearly with data. The square root means quadrupling your data only doubles your credibility — which is why a single account almost never reaches full credibility, and why two fires in five years is a single-digit-$Z$ whisper, not a verdict.
Worked example (constructed teaching example, Ch.10 figures). An account has $n = 10$ claims; the full-credibility standard is $N = 1{,}000$ claims.
SQUARE-ROOT CREDIBILITY [constructed teaching example]
n = 10 claims, N = 1,000 claims
Z = sqrt(10 / 1,000) = sqrt(0.01) = 0.10 = 10% credibility
So the estimate is 10% the account's own experience, 90% the class.
To DOUBLE that to Z = 0.20 you need FOUR times the claims (n = 40): sqrt(40/1,000) ≈ 0.20.
7. The capital behind the promises (Chapter 28)
Underwriting consumes capital, and capital is finite and costly. These numbers connect the desk to the balance sheet.
Premium-to-surplus (leverage) ratio
$$\text{Premium-to-surplus ratio} = \frac{\text{net written premium}}{\text{policyholder surplus}}$$
Meaning. A fast, crude gauge of how hard the capital is being worked. Around 1-to-1 is comfortable; the old 3-to-1 line is where supervisors traditionally grow concerned. It measures business volume, not the riskiness of the book — its blind spot is exactly the catastrophe concentration that matters most.
Worked example (constructed teaching example, Ch.28 figures). A carrier holds \$540,000,000 of surplus.
PREMIUM-TO-SURPLUS [constructed teaching example]
surplus ............ $540M
premium $540M → 540 / 540 = 1.0 : 1 conservative
premium $810M → 810 / 540 = 1.5 : 1 typical
premium $1,080M → 1,080 / 540 = 2.0 : 1 aggressive
Why it bites: at 1-to-1 a 10-point loss-ratio miss costs ~10% of surplus; at 3-to-1 the SAME miss
costs ~30%. Leverage does not change your underwriting — it multiplies the cost of getting it wrong.
Return on equity (and the cost of capital)
$$\text{Return on equity (ROE)} = \frac{\text{after-tax profit}}{\text{capital (equity)}} = \frac{\text{underwriting profit} + \text{investment income} - \text{tax}}{\text{capital consumed by the risk}}$$
Meaning. The return the capital earns for being put at risk. A risk must clear the cost of capital — the return investors could earn elsewhere at comparable risk — or it destroys value even at a combined ratio below 100%. This is the balance-sheet answer to the soft market: a catastrophe-exposed account written at a "merely adequate" combined ratio can be quietly destroying company value before any storm arrives, because it consumes far more capital per premium dollar than a benign risk.
Worked example (constructed teaching example, Ch.28 figures). Two property accounts, each \$100 of premium and each projected at a 95% combined ratio (a \$5 underwriting profit). The capital model charges the inland account \$40 of capital and the coastal account \$120 (its catastrophe tail consumes ~3× the surplus); the cost of capital is 10%.
ECONOMIC PROFIT = ROE vs. COST OF CAPITAL [constructed teaching example]
Account A (inland warehouse): UW profit $5, capital $40
capital cost = 10% × $40 = $4 → economic profit = $5 − $4 = +$1 (creates value)
Account B (coastal plant): UW profit $5, capital $120
capital cost = 10% × $120 = $12 → economic profit = $5 − $12 = −$7 (destroys value)
The combined ratio says they are identical (both 95%). The capital says A is worth writing and
B, at this price, is not — which is WHY catastrophe-exposed business must be priced to a richer margin.
8. Catastrophe: PML, AAL, and the return period (Chapter 30)
Catastrophe breaks the law of large numbers — losses are correlated, not independent — so it needs its own apparatus. These three readings come off the modeled loss distribution (the exceedance-probability curve). All figures are illustrative model outputs, not measurements.
Average annual loss (AAL)
$$\text{AAL} = \mathbb{E}[\text{annual catastrophe loss}] = \text{the mean of the whole loss distribution (the area under the EP curve)}$$
Meaning. The long-run average catastrophe loss per year — the catastrophe pure premium, the "cat load" that must be built into every rate in an exposed territory. It answers what do we charge? It is a small, smooth number; underprice it and you are underpricing the average storm, which feels free for years and then is not.
Probable maximum loss (PML)
$$\text{PML at return period } T = \text{the loss level } L \text{ such that } \Pr(\text{annual loss} > L) = \frac{1}{T}$$
Meaning. A loss read far out in the tail — the loss the portfolio would exceed only with some small, specified probability, quoted at a chosen return period (the 1-in-100-year PML, the 1-in-250-year PML). It answers a different question from the AAL: not what do we charge? but how bad is the bad case we must survive? It is the figure reinsurance and capital are sized against. Because definitions vary, always state the return period, whether it is per-event or annual, and whether it is gross or net of reinsurance.
Worked example (constructed teaching example — illustrative model output). Reading a coastal portfolio's EP curve:
PML AND AAL OFF THE EP CURVE — a coastal portfolio [constructed teaching example, illustrative $]
exceedance probability loss level return-period label
0.4% (1 / 250) ~$300M 1-in-250-year PML ← often the capital/PML standard
1.0% (1 / 100) ~$250M 1-in-100-year PML ← classic "PML" reference point
2.0% (1 / 50) ~$150M 1-in-50-year
10.0% (1 / 10) ~$30M 1-in-10-year
AAL = the mean of the WHOLE curve (the area under it) = a single, modest number — the cat load in the rate.
Read it: you size the reinsurance tower against the ~$250M–$300M PML; you load the AAL into every rate.
Return period and exceedance probability
$$\text{Return period } T = \frac{1}{\text{annual exceedance probability}}, \qquad \Pr(\text{at least one exceedance in } y \text{ years}) = 1 - (1 - \tfrac{1}{T})^{y}$$
Meaning. Two expressions of the same quantity. The exceedance probability is the annual chance a loss exceeds a level; the return period is its inverse stated as a frequency. A return period is a probability per year, with no memory — not a fixed schedule. A "1-in-100-year" event can occur twice in a decade.
Worked example (constructed teaching example). The chance a 1-in-100-year loss ($T = 100$) happens at least once over a 30-year mortgage:
RETURN PERIOD ≠ A SCHEDULE [constructed teaching example]
annual exceedance probability = 1 / 100 = 1% = 0.01
P(at least one in 30 years) = 1 − (1 − 0.01)^30 = 1 − (0.99)^30 = 1 − 0.7397 ≈ 0.26 = ~26%
Read it: a "1-in-100-year" flood has roughly a 26% chance of striking at least once during a
30-year mortgage — which is why "it won't happen in our lifetime" is a dangerous misreading.
A one-page recap
WHAT EACH NUMBER ANSWERS [reference summary]
Loss ratio (Ch.3) ............. did claims eat the premium? losses / premium
Combined ratio (Ch.3) ......... did underwriting make money? loss% + expense% (<100 = profit)
Operating ratio (Ch.3) ........ profitable after investment income? combined% − investment income%
Pure premium (Ch.6,10) ........ what will the risk cost, raw? frequency × severity
Premium build-up (Ch.11) ...... what should we charge? pure prem + expense + profit
LCM (Ch.11) ................... loss cost → a chargeable rate? loss cost × LCM
Rate indication (Ch.11) ....... is the current rate adequate? exp. LR / permissible LR − 1
X-mod (Ch.22) ................. how much does THIS employer's manual × mod
history move its price?
Schedule rating (Ch.11) ....... credit/debit for what numbers miss? manual × (1 + debits − credits)
Coinsurance (Ch.12,19) ........ what does a partial loss pay? loss × (carried/required) − deductible
ITV (Ch.15,19) ................ are we insured to rebuild cost? limit / replacement cost
Credibility blend (Ch.10) ..... trust the risk or the class? Z×own + (1−Z)×class
Square-root rule (Ch.10) ...... how much credibility does n earn? Z = sqrt(n/N)
Premium-to-surplus (Ch.28) .... how hard is the capital worked? net written premium / surplus
ROE vs cost of capital (Ch.28) does the risk earn its keep? (UW profit + inv. − tax) / capital
AAL (Ch.30) ................... the cat load to charge? mean of the loss distribution
PML at T (Ch.30) .............. the bad case to survive? loss exceeded with prob. 1/T
Return period (Ch.30) ......... how likely, and over how long? T = 1 / exceedance prob.
Keep one principle above all of them: a formula gives you a number, and a number is a starting point for judgment, not a substitute for it. The combined ratio is the truth about the past; the PML is a model's best guess about a tail; credibility tells you how little a small sample is worth. Compute the number, then — every time — read behind it. That habit, more than any single ratio on this page, is the craft.