Appendix H: Rating and Pricing Worked Examples

Appendix A gave you the formulas, each with a one-line example. This appendix does the other half of the job: it sits you down at the desk and works seven pricing problems all the way through — not a formula and a single number, but the whole arithmetic, in the order an underwriter actually does it, with the interpretation written beside every step. Because in practice the hard part was never the multiplication. It is knowing which number to reach for next, what the answer is quietly telling you, and where the model stops and your judgment starts. A price you can compute but cannot read is a price you cannot defend — to a broker pushing back, to a manager signing off, to an auditor a year later, or to yourself when the loss arrives on schedule.

The seven examples, in the order the craft builds them:

1. A pure-premium build-up to an indicated rate — from frequency × severity to a chargeable number (Ch. 10, 11).
2. An experience-rating calculation — moving the manual price by the risk's own losses (Ch. 11).
3. A schedule-rating credit/debit worksheet — pricing what the numbers can't see (Ch. 11).
4. A simplified workers'-comp X-mod illustration — how an employer's own history becomes its price (Ch. 22).
5. A coinsurance-penalty calculation — what a partial loss actually pays when the insured under-insures (Ch. 12, 19).
6. A credibility-weighting calculation — blending the risk's own experience with the class (Ch. 10).
7. A combined-ratio analysis of a small book — the scoreboard that judges all of the above (Ch. 29).

Every figure in this appendix is a constructed teaching example, marked [constructed teaching example], and is chosen to make the arithmetic legible — round, clean, and deliberately not any real insurer's experience. Real loss costs come from your carrier's filed manual, from NCCI, or from ISO/Verisk; real X-mods come from the rating bureau; real schedule-rating maxima and the exact split of a published mod come from the filed plan and the state. The worked examples teach you the shape of the calculation and how to interpret it, not what any particular account's price is. Inline math uses single-dollar delimiters; currency in the prose is escaped as \$.

One habit runs through all seven, and it is the whole point: compute the number, then read behind it.

H.1 A pure-premium build-up to an indicated rate (Ch. 10, 11)

The question on the desk. A broker sends you a light-manufacturing building and asks, "What's your rate?" Before you can answer, you have to build the rate — from the expected loss up. This is the foundational move; every later example modifies the number this one produces.

We are pricing a single class: small light-manufacturing buildings of standard construction. The exposure base is per \$1,000 of building value (Ch. 6 — the exposure unit on which the whole rating system is built). We will go from raw loss experience, to a pure premium, to loads, to an indicated manual rate — the price the math says this class should pay before we adjust for any one risk's own characteristics.

Step 1 — Frequency and severity → pure premium

Start with what the class actually costs. The pure premium is expected frequency times expected severity, and equivalently total losses divided by exposures (Ch. 10).

PURE PREMIUM = FREQUENCY × SEVERITY                   [constructed teaching example]
  Class experience (illustrative):
    4,000 buildings observed for one year ......... 4,000 building-years
    fire losses in the year ....................... 200 losses
    total incurred loss ........................... $16,000,000

  expected frequency = 200 / 4,000 .............. 0.05 losses per building-year   (a 1-in-20 chance)
  expected severity  = $16,000,000 / 200 ........ $80,000 per loss
  pure premium       = 0.05 × $80,000 ........... $4,000 per building per year
                     (check: $16,000,000 / 4,000 = $4,000 — same number, computed the other way)

Read it. The class loses, on average, \$4,000 per building per year. That is the floor — the expected loss before a single dollar of expense or profit. No rate can go below it without the insurer paying claims out of its own capital. Notice the two dimensions behave differently (Ch. 6): the 0.05 frequency is the stable, predictable number; the \$80,000 severity is an average that hides a long right tail, where one total-loss fire could be ten times the mean. We price off the average here, but we never forget the tail — that is what the catastrophe load and the per-risk limit are for.

Step 2 — Convert to the exposure base

The rate is quoted per \$1,000 of value, so put the pure premium on that base. For a representative \$1,000,000 building:

PURE PREMIUM PER $1,000 OF VALUE                      [constructed teaching example]
  pure premium per building ...... $4,000
  building value ................. $1,000,000  →  1,000 units of $1,000
  pure premium per $1,000 = $4,000 / 1,000 = $4.00 per $1,000 of value

Read it. \$4.00 per \$1,000 is the expected-loss component of the rate. The rating bureaus publish exactly this quantity by class and call it the loss cost (Ch. 10) — it is the pure premium under a different name.

Step 3 — Add the loads (expense, then profit and contingencies)

A rate that stops at the pure premium is not a rate; it is a subsidy. The carrier has to pay commission, run the underwriting and issuance operation, pay premium taxes, and earn a margin (with a cushion for being wrong). Stack the three blocks (Ch. 11). Here the carrier's expenses run about 30% of premium and its target profit and contingencies about 5% of premium — so the pure premium must be 65% of the final rate (the permissible loss ratio: $1 - 0.30 - 0.05 = 0.65$).

PREMIUM BUILD-UP ($ per $1,000 of building value)     [constructed teaching example]
  pure premium (expected loss)        ████████████████  $4.00   ← from Step 2
  + expense load (~30% of premium)    ███████           $1.85   ← commission, ops, taxes, overhead
  + profit & contingencies (~5%)      ██                $0.40   ← margin + cushion for being wrong
  ───────────────────────────────────────────────────
  = indicated manual rate                               $6.25 per $1,000   (before any risk modification)

A note on the arithmetic, because loads are where new underwriters slip. The loads are a percentage of the final premium, not of the pure premium — you are grossing the \$4.00 up to a number of which it is 65%. The clean way: indicated rate $= \$4.00 / 0.65 = \$6.15$; the build-up above shows \$6.25 because the 30% and 5% are rounded display figures rather than exact reciprocals. Either way the lesson is the same — the expense and profit blocks roughly half-again the pure premium, which is why quoting the loss cost as the rate (a classic soft-market error) sells coverage with no expense load and no profit at all.

Step 4 — Apply it to the building

INDICATED PREMIUM FOR A $20,000,000 BUILDING          [constructed teaching example]
  indicated manual rate ...... $6.25 per $1,000
  building value ............. $20,000,000  →  20,000 units of $1,000
  indicated building premium = $6.25 × 20,000 = $125,000

What this number is, and what it is not. \$125,000 is the manual indication: the price for an average-quality member of this class. It is the starting line, not the finish. It does not yet know that this building has a thirty-year-old roof, or a sprinkler system, or a clean ten-year loss run, or a plant manager who runs a tight hot-work program. Those move the price through experience and schedule rating — Examples H.2 and H.3, which take this exact \$125,000 forward.

Underwriter's takeaway. The build-up answers "what should an average risk in this class pay?" Memorize the three blocks — expected loss, the cost of doing business, the margin — because every pricing conversation you ever have is a negotiation about one of them. And hold the discipline of Chapter 11: a rate that drops a block to win the account does not become profitable; it becomes a loss that shows up in two or three years, after the broker has thanked you and moved on.

H.2 An experience-rating calculation (Ch. 11)

The question on the desk. The manual says \$125,000. But this insured has its own five-year loss history. Is that history good enough to move the price — and by how much, given that five years of one mid-size account is a small sample?

Experience rating adjusts the manual premium up or down by the risk's own losses relative to what its class would predict — weighted by how credible that experience is (Ch. 10, 11). It is retrospective and formulaic. The mechanics differ by line and bureau; what follows is the underlying logic, which is the same everywhere: compare actual to expected, then trust the comparison only as far as the data earns.

Step 1 — Actual vs. expected losses

EXPERIENCE RATIO — ACTUAL vs. EXPECTED                [constructed teaching example]
  Account's own experience period: 5 years
    actual incurred losses (trended & developed, Ch.10) ... $385,000
  Expected losses for an account this size in this class:
    expected loss rate × the account's 5-year exposure .... $550,000

  experience ratio = actual / expected = $385,000 / $550,000 = 0.70

Read it. This account ran losses 30% below what its class would have predicted over the same period. On its face that argues for a credit — a price below manual. But "on its face" is exactly where a careful underwriter slows down. Two questions before we trust it: are these losses developed (have the open claims been grown toward their ultimate value, Ch. 10) and trended to today's cost level? And — the harder one — is five years of one account enough data to believe? A 0.70 from a sample of three small claims is a different animal from a 0.70 from a long, dense, stable record.

Step 2 — Apply credibility to the indicated swing

We do not hand the account the full 30% credit. We weight the experience by its credibility $Z$ (Ch. 10), blending the indicated modification toward 1.00 (no change). Here we assess $Z = 0.40$ — a partially credible record, typical for a single mid-market account over five years.

CREDIBILITY-WEIGHTED MODIFICATION                     [constructed teaching example]
  raw indicated mod (from the experience ratio) .... 0.70   (a 30% indicated credit)
  credibility ...................................... Z = 0.40
  experience modification = Z × (experience ratio) + (1 − Z) × 1.00
                          = 0.40 × 0.70 + 0.60 × 1.00
                          = 0.28 + 0.60
                          = 0.88

Read it. The account's good history earns it a credit — but only 12%, not 30%. At 40% credibility the account's own experience gets 40% of the vote and the class default (1.00, "you're average until proven otherwise") gets 60%. Credibility is the humility the math forces on you: the better-than-class result is real enough to reward, not real enough to bet the full amount on. (Had the indicated ratio been above 1.00 — a worse-than-class history — the identical formula would have produced a debit, capped the same way toward 1.00.)

Step 3 — Apply the mod to the manual premium

EXPERIENCE-RATED PREMIUM                              [constructed teaching example]
  manual premium (from H.1) ...... $125,000
  × experience modification ...... × 0.88
  = experience-rated premium ..... $110,000
  ($15,000 less than manual — the credit the account's own losses earned, after credibility)

Underwriter's takeaway. Experience rating is the formula's verdict on the past. It rewards (or punishes) the account for its own track record, but only to the extent the record is credible — which for a single account is almost never "fully." The trap (Ch. 11) is granting the full indicated credit on a thin record because the broker frames the losses as "clean." Clean and thin still earns only partial credibility. We carry the \$110,000 forward to H.3, where judgment — not the formula — gets its say about the future.

H.3 A schedule-rating credit/debit worksheet (Ch. 11)

The question on the desk. The formula priced the past (H.2). But the loss run cannot see the new roof the insured just contracted, the sprinkler upgrade, the housekeeping that an inspection flags, or the safety culture you can feel the moment you walk the plant. Schedule rating is where the underwriter prices those — the things the numbers miss.

Schedule rating applies documented credits (reductions) and debits (increases) to the premium for risk characteristics not captured by the class rate or by experience rating: management quality, premises and protection, equipment and maintenance, and loss-control programs (Ch. 11). It is prospective and judgmental — the mirror image of experience rating's retrospective formula. Most filed plans cap the net swing (commonly around ±25% — treat the exact cap as filing- and state-specific).

Step 1 — Itemize each credit and debit, with the reason

The cardinal rule: every modification needs a documented reason in the file. A credit you cannot defend in writing is a "soft credit," and soft credits are how a disciplined book gets underpriced one reasonable-looking concession at a time.

SCHEDULE-RATING WORKSHEET                             [constructed teaching example]
  Category                         Credit / Debit   Documented reason (goes in the file)
  ──────────────────────────────────────────────────────────────────────────────────────
  Management & cooperation          −5%   (credit)  Experienced owner; tight ops; responsive to loss control
  Premises & protection (roof)     +10%   (debit)   Roof at end of service life — near-term wind/water driver
  Premises & protection (housekp.)  +5%   (debit)   Combustible scrap allowed to accumulate near hot work
  Protective devices (sprinklers)   −5%   (credit)  Wet-pipe system upgraded and recently inspected
  Loss-control / safety program     −5%   (credit)  Verified hot-work permit program now in place
  ──────────────────────────────────────────────────────────────────────────────────────
  Net schedule modification = (−5) + (+10) + (+5) + (−5) + (−5) = 0%

Read it. The debits and credits very nearly cancel — this risk is, on the schedule, roughly average once you net its strengths against its weaknesses. That is itself a finding: the new safety program and sprinkler upgrade are pulling against the aging roof and the housekeeping problem. The net came to 0%, but do not mistake "zero net" for "nothing to do." The +10% roof debit is a live exposure; the −5% safety credit is contingent on the program actually being in force.

Step 2 — Apply the net modification to the premium

SCHEDULE-RATED PREMIUM                               [constructed teaching example]
  premium before schedule rating (from H.2) ... $110,000
  × (1 + net schedule modification) ........... × (1 + 0.00) = × 1.00
  = schedule-rated premium .................... $110,000

In this illustration the net is zero, so the number does not move — but watch how sensitive it is. Suppose the verified safety credit did not exist yet (the program is "planned," not in place). Drop that −5% and the net becomes +5%:

SAME RISK, ONE UNVERIFIED CREDIT REMOVED             [constructed teaching example]
  net schedule modification = (−5) + (+10) + (+5) + (−5) = +5%   (a net DEBIT)
  schedule-rated premium = $110,000 × 1.05 = $115,500

Read it. A single \$5,500 swing turns entirely on whether one control is verified or merely promised. This is the discipline of Chapter 11 made concrete: you credit a control you can document and you do not credit one you cannot. The honest move when the program is "planned" is not to grant the credit and hope — it is to make the program a subjectivity (Ch. 13): bind on condition that the documented hot-work program is in place within, say, sixty days, and grant the credit at renewal once it is verified.

Underwriter's takeaway. Experience rating (H.2) priced what the account did; schedule rating prices what the account is and will do. Together they pull the manual rate toward the individual risk from two directions — formula and judgment. A risk can carry an experience credit for a clean past and a schedule debit for an aging roof with no contradiction at all; they answer different questions. The number to distrust is always the credit you took on faith.

H.4 A simplified workers'-comp X-mod illustration (Ch. 22)

The question on the desk. Workers' comp is statutory — the benefits are set by law, not by the policy, and the core coverage carries no dollar limit (Ch. 22). So how does the carrier price one employer above or below another? Through the experience modification factor — the X-mod — a single filed multiplier that turns the employer's own loss history into its price.

The published X-mod (NCCI or a state bureau) is more intricate than what follows — it splits every claim into a heavily weighted primary portion and a lightly weighted excess portion, and it folds in credibility-style ballast and weighting values from the filed tables. You do not compute the official mod; you read it off the bureau's worksheet and you read behind it. But the simplified version below shows the engine clearly, and above all shows the one feature every underwriter must internalize: the mod responds far more to claim frequency than to severity.

Step 1 — Actual vs. expected losses, primary vs. excess

The mechanism that makes frequency dominate: each claim's losses are split at a primary/excess threshold (here, illustratively, \$15,000). The *primary* portion (the first \$15,000 of every claim) is weighted heavily; the excess portion (everything above \$15,000) is discounted hard. So many small claims pile up primary dollars — the part the formula listens to — while one large claim contributes one capped primary slice and a heavily discounted excess remainder.

Consider two welding shops with the same total incurred losses and the same expected losses, differing only in the shape of their loss history.

TWO SHOPS, SAME TOTAL LOSSES, DIFFERENT SHAPE        [constructed teaching example]
  Primary/excess split threshold (illustrative) ... $15,000 per claim
  Expected losses for each shop's size/class ...... $150,000  (primary-equivalent, illustrative)

  SHOP A — one large claim
    Claims: 1 claim of $120,000
    primary = min($120,000, $15,000) ............... $15,000
    excess  = $120,000 − $15,000 ................... $105,000  (heavily discounted in the formula)

  SHOP B — six small claims
    Claims: 6 claims of $20,000 each (= $120,000 total)
    primary = 6 × min($20,000, $15,000) = 6 × $15,000 = $90,000
    excess  = 6 × ($20,000 − $15,000)   = 6 × $5,000  = $30,000

Read it before computing anything. Both shops lost \$120,000. But Shop A generated only **\$15,000 of primary loss, while Shop B generated \$90,000 — six times as much. The formula is built to treat primary dollars as the honest signal of how often this workplace hurts people, because frequency is the stable, predictable, controllable dimension (Ch. 6) and a reliable predictor of future loss. One \$120,000 fluke says less about next year than six \$20,000 injuries do.

Step 2 — Form the (simplified) modification

A simplified mod compares weighted actual losses to expected losses; primary losses enter at full weight, excess at a heavily reduced weight (here, illustratively, 10%).

SIMPLIFIED X-MOD (illustrative weighting)            [constructed teaching example]
  Simplified mod ≈ [primary actual + 0.10 × excess actual] / expected losses

  SHOP A: [$15,000 + 0.10 × $105,000] / $150,000
        = [$15,000 + $10,500] / $150,000
        = $25,500 / $150,000 = 0.17 → mod ≈ 0.85   (a CREDIT mod; better-than-class)

  SHOP B: [$90,000 + 0.10 × $30,000] / $150,000
        = [$90,000 + $3,000] / $150,000
        = $93,000 / $150,000 = 0.62 → mod ≈ 1.25   (a DEBIT mod; worse-than-class)

(The step from the raw ratio to the published 0.85 / 1.25 runs through the bureau's ballast and weighting values, which stabilize small accounts and which we are not reproducing; the figures here are illustrative endpoints chosen to show the direction and rough magnitude, not a filed calculation.)

Read it. Same \$120,000 of losses, and the mod swings from a 15% credit to a 25% debit purely on the frequency pattern. This is the single most important fact about workers' comp pricing, and it is counterintuitive to newcomers: the shop with the one big claim is priced better than the shop with six small ones. The formula is telling you that a workplace which keeps hurting people — even in small ways — is the worse forward risk.

Step 3 — Apply the mod to manual premium

The X-mod multiplies the manual premium (itself the sum of each NCCI class code's loss cost × that code's payroll per \$100, Ch. 22).

THE X-MOD APPLIED                                    [constructed teaching example]
  manual premium (class rates × payroll) ...... $200,000
  Shop A: $200,000 × 0.85 = $170,000   (a $30,000 credit — rewarded for low frequency)
  Shop B: $200,000 × 1.25 = $250,000   (a $50,000 surcharge — penalized for high frequency)

Underwriter's takeaway. The X-mod is the purest example in the book of an account's own behavior becoming its price (Ch. 22). You do not invent it — you read it — but you must understand what it is saying: frequency, not severity, moves the mod, so a rising mod driven by many small claims is a red flag about the management of safety, not just bad luck. And it points straight at the underwriter's best loss-control lever — a return-to-work program shortens claim duration and cuts the indemnity dollars feeding next year's mod. The mod is not just a price; it is a diagnosis.

H.5 A coinsurance-penalty calculation (Ch. 12, 19)

The question on the desk. The insured carries \$6M on a building and suffers a \$2M fire — well within the limit. They expect a \$2M check (less the deductible). They will not get it. Why, and how much short?

Coinsurance is a terms problem, not a price problem — a property condition that punishes under-insurance (Ch. 12). It exists to force insurance-to-value and to defeat the adverse-selection move of buying only enough limit to cover the partial losses one expects to have (Ch. 19). If the insured carries less than the required percentage (commonly 80/90/100%) of the property's full value, the insurer pays only a proportional share of every loss — even a partial one far below the limit.

The formula (Ch. 12):

$$\text{Loss payment} = \text{loss} \times \frac{\text{insurance carried}}{\text{insurance required}} - \text{deductible}$$

where insurance required $=$ coinsurance percentage $\times$ the property's full value at the time of loss, and the fraction is capped at 1 (carrying more than required does not over-pay a partial loss).

Step 1 — Establish the required amount

INSURANCE REQUIRED — an 80% coinsurance clause        [constructed teaching example]
  full value at time of loss ........ $10,000,000
  coinsurance percentage ............ 80%
  insurance required = 0.80 × $10,000,000 = $8,000,000

The critical, often-missed point: the test uses the value at the time of loss, not the value when the policy was written. A year of construction-cost inflation that quietly raises full value while the limit stays flat erodes the ratio and deepens the penalty (Ch. 19 — insurance-to-value is a moving target).

Step 2 — Form the penalty ratio

PENALTY RATIO = CARRIED / REQUIRED                   [constructed teaching example]
  insurance carried ........ $6,000,000
  insurance required ....... $8,000,000
  penalty ratio = $6,000,000 / $8,000,000 = 0.75

Read it. The insured carried only 75% of what the clause required, so the policy will pay only 75 cents on the dollar of any loss — including this partial one. The insured has, without realizing it, made themselves a coinsurer of their own loss for the missing 25%.

Step 3 — Apply to the loss, then the deductible

THE COINSURANCE PENALTY                              [constructed teaching example]
  partial loss ............. $2,000,000
  × penalty ratio .......... × 0.75
  = before deductible ...... $1,500,000
  − deductible ............. $50,000
  = loss payment ........... $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 = the coinsurance penalty ($500,000) plus the deductible ($50,000).
  The LIMIT was never the constraint; the VALUE was.

Step 4 — The contrast that teaches it

Show the same loss with the insured properly insured to value (\$8M carried, meeting the 80% requirement):

SAME LOSS, PROPERLY INSURED TO VALUE                 [constructed teaching example]
  carried $8,000,000 ≥ required $8,000,000  →  penalty ratio = capped at 1.00 (no penalty)
  loss payment = $2,000,000 × 1.00 − $50,000 = $1,950,000
  The difference — $1,950,000 vs. $1,450,000 = $500,000 — is the price of under-insuring.

Underwriter's takeaway. Coinsurance is where structuring (Ch. 12) and valuation (Ch. 19) meet a partial loss and the insured discovers the policy means exactly what it says. For the underwriter, chronic under-insurance is not only the insured's claims problem — it is your premium problem, because you collected rate on \$6M of value all year when \$8M was at risk. This is why agreed value exists (Ch. 19): the insurer and insured fix the value in advance against a signed statement of values, the underwriter verifies it, and the coinsurance clause is suspended for the term — certainty in exchange for verified confidence in the number. Verify the value; do not discover it at the claim.

H.6 A credibility-weighting calculation blending risk and class experience (Ch. 10)

The question on the desk. An account's own loss ratio looks alarming — 95% against a class that runs 62%. Do you price to the account's frightening history, or to the comfortable class average? Neither. You price to a blend, and credibility theory tells you the mixing ratio.

This is the single most important idea in the math of underwriting, and the one newcomers get wrong most often — in both directions. They over-react to a small account's bad year (pricing to the 95% as if it were destiny), or they ignore a genuine signal (waving the 95% away as noise). Credibility (Ch. 10) is the disciplined middle: listen to the account's own experience exactly as much as it deserves, and let the class supply the rest.

The formula:

$$\text{Estimate} = Z \times (\text{risk's own experience}) + (1 - Z) \times (\text{class expectation})$$

Step 1 — Assemble the two inputs and the weight

THE THREE INGREDIENTS                                [constructed teaching example]
  risk's own loss ratio (trended, developed, Ch.10) ... 95%   (looks like a serious problem)
  class loss ratio (the benchmark) .................... 62%   (the default if we knew nothing else)
  credibility of the account's own data ............... Z = 0.30

Where did $Z = 0.30$ come from? From the volume of the account's own claim experience, not its direction (Ch. 10). Under the classical square-root rule, $Z = \sqrt{n/N}$, where $n$ is the claims observed and $N$ is the full-credibility standard. If this account produced about $n = 90$ claims against a full-credibility standard of $N = 1{,}000$:

WHERE Z COMES FROM (square-root rule)                [constructed teaching example]
  Z = sqrt(n / N) = sqrt(90 / 1,000) = sqrt(0.09) = 0.30
  Read it: credibility grows with the SQUARE ROOT of data, so to double Z to 0.60 you would need
  FOUR times the claims (n = 360). A single account almost never reaches full credibility — which is
  the whole reason we blend.

Step 2 — Blend

CREDIBILITY-WEIGHTED LOSS RATIO                      [constructed teaching example]
  estimate = Z × own + (1 − Z) × class
           = 0.30 × 95% + 0.70 × 62%
           = 28.5% + 43.4%
           = 71.9% ≈ 72%

Read it. The account's bad history pulls the estimate up from the 62% class — but only to 72%, not 95%. At 30% credibility the account's own small-sample experience earns 30% of the vote and the class earns 70%. The 95% was real enough to move the number 10 points above the class; it was not real enough to be believed on its own. That gap — between what a single bad year looks like and what it predicts — is exactly where undisciplined pricing lives.

Step 3 — Turn the blended loss ratio into a price signal

A credibility-weighted loss ratio is not yet a rate; it is the input to a rate indication (Ch. 11). Compare it to the permissible loss ratio — the share of premium the rate allows for losses after expenses and profit. With expenses of 30% and profit/contingencies of 5%, the permissible loss ratio is $1 - 0.35 = 65\%$.

FROM BLENDED LOSS RATIO TO RATE INDICATION           [constructed teaching example]
  credibility-weighted loss ratio .... 72%
  permissible loss ratio ............. 65%
  indicated rate change = 72% / 65% − 1 = 1.108 − 1 = +10.8%

Read it. Priced to the blended experience, this account needs roughly an 11% increase to reach adequacy. Priced naively to its own raw 95% you would have demanded an unsupportable ~46% increase and lost a fundamentally average account to a competitor; priced lazily to the 62% class you would have under-charged a risk that is genuinely running a little hot and booked the shortfall as a loss two years out. Credibility is what keeps you off both rocks.

Underwriter's takeaway. Credibility is humility expressed as arithmetic (Ch. 10). The blend is not a hedge or a split-the-difference cop-out — it is the mathematically defensible estimate, and a number you can show an auditor. When a broker waves a single great year to demand a deep credit, or a single bad year as proof an account is uninsurable, the disciplined answer is the same: how credible is that experience? Then weight it accordingly, and price to the blend.

H.7 A combined-ratio analysis of a small book of business (Ch. 29)

The question on the desk. You have written a few dozen accounts using everything above. Now the question is no longer "is this risk any good?" but "is this book any good?" — because underwriting is portfolio construction, not just risk selection (Ch. 29). The combined ratio is the scoreboard, and it tells the truth.

A book has properties no single account has — a blended loss ratio, an expense ratio, a mix, a set of concentrations, and a trajectory. We will take a small illustrative book, segment it (Ch. 29), and read what the average hides.

Step 1 — The book in total

The combined ratio is the loss ratio plus the expense ratio: the share of every premium dollar paid out in losses and expenses (Ch. 3). Below 100% is an underwriting profit; above 100% is an underwriting loss, before any investment income.

THE WHOLE BOOK                                       [constructed teaching example]
  earned premium .................................... $10,000,000
  incurred losses + loss adjustment expense ......... $6,500,000
  underwriting expenses (acquisition + general) ..... $3,000,000

  loss ratio     = $6,500,000 / $10,000,000 = 65%
  expense ratio  = $3,000,000 / $10,000,000 = 30%
  combined ratio = 65% + 30% = 95%

Read it. A 95% combined ratio is a 5-point underwriting profit — the book keeps about 5 cents of every premium dollar on underwriting alone, before a dollar of investment income. On the surface, a healthy commercial result. But the book-level average is exactly where the dangerous problems hide (Ch. 29). A single number cannot tell you whether a profitable core is quietly subsidizing a segment that is bleeding. So we segment.

Step 2 — Segment the book

Slice the same \$10M along a meaningful dimension — here, industry segment — and compute each slice's loss ratio (Ch. 29). Expenses run ~30% across the board, so we can read profitability straight off the loss ratios against the 65% permissible loss ratio (the break-even-on-underwriting line for a 5-point profit target).

SEGMENT CHECK — does it reconcile?                   [constructed teaching example]
  premium:  4.0 + 2.5 + 2.0 + 1.5 = $10.0M   ✓
  loss+LAE: 2.2 + 1.5 + 1.4 + 1.4 = $6.5M    ✓  → blended loss ratio 6.5/10.0 = 65%   ✓

Read it. The 95% average was telling the truth and hiding the story at the same time. Light manufacturing (85%) is carrying the book; coastal property is running a 123% combined ratio — a 23-point underwriting loss — and contractors sits right on the 100% waterline. The profitable core is subsidizing the coastal segment. The average looked fine precisely because the strong segments are large enough to mask the weak one.

Step 3 — Read the concentration, not just the loss ratio

A loss ratio tells you a segment's past result. Concentration tells you its capacity to hurt you in one event (Ch. 29). The coastal segment is not merely unprofitable — it is correlated: its accounts share the same named-storm peril, so one hurricane can hit many of them at once, and the independence the law of large numbers assumes fails exactly when it is needed.

CONCENTRATION READ — coastal property segment        [constructed teaching example]
  Coastal premium ......................... $1,500,000  (15% of the book by premium)
  Accounts in the same peril zone ......... clustered — a single storm strikes many at once
  The 123% is the AVERAGE year. The bad year (the storm year) is far worse, and arrives correlated.
  → This segment's true cost is not its loss ratio; it is its contribution to the book's tail (Ch.30).

Read it. Two separate problems are stacked here, and a junior underwriter sees only the first. The segment is underpriced (the 123% says the rate is inadequate for the average year). And it is over-concentrated (one event can convert a bad segment into a solvency event). The discipline of Chapter 29 says the fix is not one lever but a choice among several: raise rate to adequacy, tighten terms (higher named-windstorm deductibles, Ch. 12), cap the segment's growth, or cede more of it to reinsurance (Ch. 27) — and, above all, manage the accumulation, because the value of any one coastal account to the book depends entirely on how much coastal exposure is already in it.

Step 4 — What the analysis tells you to do

THE BOOK-MANAGEMENT VERDICT                          [constructed teaching example]
  KEEP & GROW   Light manufacturing (85%) ...... the profit engine; feed it, defend renewals
  HOLD          Habitational (90%) ............... solid; monitor
  WATCH/RE-PRICE Contractors (100%) .............. on the waterline; needs rate or tighter selection
  FIX or SHED   Coastal property (123%) ......... raise rate, tighten terms, cap growth, cede more,
                                                   and manage the accumulation — NOT just "write less"

Underwriter's takeaway. The combined ratio is the truth — but only when you read it at the right altitude (Ch. 29). At the book level, 95% looked like success; segmented, it revealed a profitable core subsidizing a concentrated, underpriced coastal segment that one storm could turn into a crisis. This is the portfolio lesson the whole book builds toward: a stack of individually defensible accounts can still be a badly built book, and the manager's job is to see the correlation, the concentration, and the subsidy that no single file shows. Compute the average; then segment it; then act on the part, not the whole.

A note on putting these together

These seven examples are not seven separate skills — they are one pricing-and-portfolio pipeline, and in practice you run them in sequence on a real account:

THE PIPELINE                                         [reference summary]
  H.1  Build the manual rate ............ pure premium + loads → indicated price   (Ch. 10, 11)
  H.2  Modify for the past .............. experience rating, credibility-weighted   (Ch. 11)
  H.3  Modify for the rest .............. schedule rating credits/debits, documented (Ch. 11)
  H.4  Price comp by its own history .... the X-mod (frequency-weighted)            (Ch. 22)
  H.5  Structure against under-insurance  coinsurance / insurance-to-value          (Ch. 12, 19)
  H.6  Trust the data correctly ........  credibility blend of risk vs. class       (Ch. 10)
  H.7  Judge the book, not just the risk  combined-ratio + concentration by segment (Ch. 29)

Notice that the same disciplines surface again and again across the seven: credibility governs both the experience mod (H.2) and the loss-ratio blend (H.6); frequency over severity explains both the pure premium's stable first dimension (H.1) and the X-mod's verdict (H.4); document-before-you-credit runs from the schedule worksheet (H.3) to the verified value behind a coinsurance waiver (H.5); and the combined ratio that judges the whole book (H.7) is just the loss and expense loads of H.1 played out across a hundred accounts and a few years.

And the one habit that runs under all of it, the habit Appendix A ended on and this appendix has shown at work seven times: a formula gives you a number, and a number is the start of an underwriting judgment, not a substitute for it. The build-up gives a rate; credibility tells you how far to trust the history behind it; the X-mod gives a multiplier and a diagnosis of a workplace; the combined ratio gives a verdict that changes meaning the moment you segment it. Compute the number. Then — every single time — read behind it. That habit, more than any worksheet on these pages, is the craft.