Chapter 10 Exercises

Work these the way you would work a real loss run: keep frequency and severity apart, ask whether a number is trended and developed before you trust it, and run the credibility math before you let a handful of claims move your price. Items marked with a dagger (†) have worked solutions in Appendix: Answers to Selected Exercises; the rest are for discussion or self-test. Section references like (§10.5) point you back to the relevant part of the chapter. Treat every dollar figure as a constructed teaching number.

A. Recall and definitions

  1. † Define pure premium two equivalent ways — once as frequency × severity, once as a ratio of totals — and explain why the two are the same. (§10.3)
  2. State the loss ratio in its honest form. Which premium figure belongs in the denominator and which loss figure in the numerator, and why does using the other forms mislead? (§10.2)
  3. In one sentence each, distinguish trend from development. Which one corrects for inflation-like change over time, and which for the immaturity of open and not-yet-reported claims? (§10.4)
  4. † Define credibility ($Z$), full credibility, and partial credibility. What does $Z = 0$ instruct you to do, and what does $Z = 1$ instruct you to do? (§10.5)
  5. Write the credibility-weighting formula from memory and explain, in words, what each of its two terms contributes. (§10.6)
  6. Why is a frequency distribution described as "low-count and lumpy" while a severity distribution is described as "skewed with a long right tail"? What does the skew do to the relationship between the mean and the median claim? (§10.1)
  7. Define the exposure base and give the standard base for (a) commercial property, (b) workers' compensation, (c) commercial auto. What two properties make a base a good one? (§10.3)

B. Frequency, severity, and the shape of loss

  1. † A class of shops has expected frequency 0.5 claims per year and expected severity \$120,000 per claim. (a) What is the expected annual loss? (b) If the exposure base is \$1,000 of building value and the class carries \$300,000,000 of value across 300,000 units, what is the pure premium per unit, given the same \$X total losses you compute in (a) scaled to the class? (State any assumption you make.) (§10.1, §10.3)
  2. Account A: many small claims, no large ones. Account B: years of zeros punctuated by one seven-figure loss. Both cost the same expected dollars per year. Classify each as a frequency risk or a severity risk, and say which one a higher deductible helps and which one a policy limit / reinsurance protects. (§10.1)
  3. † A property loss run shows claims of \$3K, \$5K, \$8K, \$12K, \$22K, and \$1,150,000. Compute the median and the mean claim. Explain to a trainee why quoting "the average claim is \$200,000" would badly misrepresent the typical loss — and why ignoring the \$1.15M would be just as wrong. (§10.1)
  4. A welding shop had zero property claims last year. Using the idea that frequency is a low-count distribution, explain why one clean year is weak evidence the risk is good — and what you would look at instead. (§10.1)

C. Loss ratio

  1. † An account brought in \$2,000,000 of *written* premium; \$1,500,000 of it is earned. Incurred losses are \$975,000; paid losses are \$500,000. Compute the loss ratio four ways (paid/written, paid/earned, incurred/written, incurred/earned). Which is the honest figure, and what is it? (§10.2)
  2. Your insurer's expenses run about 28% of premium and it targets 6% profit and contingencies. What is the permissible (target) loss ratio, and what does it mean to write an account you expect to run above it? (§10.2)
  3. † A new workers'-comp book reports a 35% paid loss ratio in its first year, and the sales team wants to triple it. Explain, using development and earned vs. written premium, why this number is almost certainly too good — and what figure you would judge the book on instead. (§10.2, §10.4)
  4. Why is a single account's one-year loss ratio nearly meaningless, while a book's loss ratio or a single account's multi-year loss ratio is far more useful? Connect your answer to credibility. (§10.2, §10.5)

D. Pure premium, trend, and development

  1. † A class produced \$9,000,000 of incurred losses across 600,000 exposure units. (a) Compute the pure premium per unit. (b) If the permissible loss ratio is 65%, estimate the charged rate per unit before any credits or debits. (§10.3)
  2. A liability claim from three years ago is valued today at \$300,000 and is still open. Apply an illustrative loss development factor of 1.30 and then trend the developed figure forward three years at 5% per year. What is the trended, developed (ultimate, future-cost) loss, and by what percentage does it exceed the raw \$300,000? (§10.4)
  3. † Explain why a recent accident year is simultaneously the most relevant and the least reliable year of experience for pricing. What does this imply about how you weight recent versus older years? (§10.4)
  4. Match the line to how much development you should expect, and say why: (a) commercial property fire; (b) general liability; (c) workers' compensation. Then say which of the three needs the most attention to trend as well. (§10.4)

E. Credibility — the heart of the chapter

  1. † Using the square-root rule with a full-credibility standard of $N = 1{,}000$ claims, compute $Z$ for an account with (a) 10 claims, (b) 90 claims, (c) 250 claims. State in words what "quadrupling the data only doubles the credibility" means. (§10.5)
  2. An account's own trended/developed loss ratio is 105%; its class runs 68%; you assess $Z = 0.20$. Compute the credibility-weighted loss ratio. Is the account's own (bad) experience earning a large or a small share of the answer, and why? (§10.6)
  3. † Explain the two opposite errors credibility weighting protects against — over-reacting to a risk's own bad experience, and ignoring a risk's own experience entirely. Give a one-line example of each from a line you know. (§10.6)
  4. In plain English, state the Bühlmann question — "is the difference I'm seeing signal or noise?" — and say what makes credibility high (which variance large, which small) and what makes it low. You do not need the formula. (§10.7)
  5. Why does a life-insurance applicant like David Okafor have essentially zero own-experience credibility for the event being priced, and what does the underwriter rely on instead? What general principle about classification does this illustrate? (§10.6)

F. Underwrite this submission

  1. † Underwrite the credibility call. A mid-size commercial account comes in with a 5-year loss history: one large fire (\$900K) and three small claims (\$8K, \$15K, \$20K). The class loss ratio is 60%; the account's own raw loss ratio over the period is 92%. You judge $Z = 0.25$. (a) Compute the credibility-weighted loss ratio. (b) State the defensible expected loss ratio you would carry into pricing. (c) Name the one qualitative finding that would make you override the blend upward anyway. (§10.6, §10.1)
  2. Read the small history. An account has had exactly one claim in four years — a \$2,000,000 liability loss. The broker says "one claim in four years, that's a great risk, give them a credit." Write the two-sentence reply that uses both frequency-as-a-distribution and severity-as-a-tail to explain why one claim of that size is not, by itself, evidence of a good risk. (§10.1, §10.5)

G. Price this risk (calculation)

  1. † Build a pure premium and a rate. A class of small machine shops generated \$4,500,000 of trended, developed incurred losses across 250,000 units of \$1,000 payroll. (a) Compute the pure premium (loss cost) per unit. (b) With a 65% permissible loss ratio, compute the indicated charged rate per unit before credits/debits. (c) A shop in this class has 1,000 units of payroll; what is its indicated manual premium before any experience or schedule adjustment? (§10.3, §10.2)
  2. Blend, then price. An account's own pure premium indication is \$22 per exposure unit; the class loss cost is \$15. You assess $Z = 0.40$. (a) Compute the credibility-weighted pure premium. (b) At a 65% permissible loss ratio, what charged rate per unit does the blend imply? (§10.3, §10.6)
  3. † The trend you forgot. You priced an account two years ago off losses that you did not trend. Severity has run about 6% per year since. Roughly how much has your original pure premium fallen behind the true current cost, and what does that do to your loss ratio if your premium did not move? (§10.4, §10.2)

H. Find the red flag

  1. A broker's submission includes a loss summary showing a 38% loss ratio "based on losses paid to date" for a three-year-old book of contractor general-liability business. Identify the red flag and state the two questions you must ask before you trust the 38%. (§10.2, §10.4)
  2. † An underwriter prices a mid-size account entirely on its own two-year loss experience, ignoring the class, because "the account's own numbers are right there." Name the statistical error, estimate the kind of credibility two years of a mid-size account actually carries, and say what the underwriter should have done. (§10.5, §10.6)
  3. A rating model assigns a catastrophe-exposed coastal property account high credibility on its own five-year loss experience (which happens to be clean). Using the Bühlmann intuition, explain why that high credibility is suspicious for this line. (§10.7)

I. Memo and communication

  1. † Write the memo. In 150–200 words, explain to your underwriting manager why you are not pricing Harbor Steel off its raw two-fire loss history. Walk through: frequency vs. severity, the low credibility of two claims, the credibility-weighting toward the class, and the one exception (the hot-work severity signal) that you are handling through terms rather than through the base rate. (§10.1, §10.5, §10.6, The Underwriting File)
  2. A trainee asks, "If the credibility math says shrink toward the class, why do we even bother reading the loss run in detail?" Write a short, concrete answer that defends both the math and the close reading. (§10.6)

J. Ethics and judgment

  1. Ethics dilemma. Your insurer's credibility-weighted indication for a small minority-owned business in a high-crime ZIP code comes out high, driven mostly by the class/territory loss cost rather than the account's own (clean) experience. The owner protests that they are being punished for their neighborhood, not their risk. Lay out the genuine tension between actuarial fairness (the class loss cost is real) and the fairness concern the owner is raising, and name where the legal line on fair vs. unfair discrimination sits (cite the owning chapters). Do not resolve it glibly. (§10.6; fair vs. unfair discrimination, §4.7 / Ch. 35)
  2. An underwriter, under pressure to hit a growth target in a soft market, quietly stops trending and developing the loss runs on new submissions because the raw numbers produce lower, more competitive premiums. Explain why this is both a discipline failure and an eventual combined-ratio failure, and what it has in common with adverse selection. (§10.4, §10.2; §1.4)

K. The Underwriting File (extensions)

  1. † Extend the file. Using only the chapter's methods and the frozen Harbor Steel facts, write the two-sided disposition this chapter is supposed to record: (a) the math conclusion about how much the two fires should move the price, and (b) the judgment conclusion about the one part of that history the credibility math would underweight. Be explicit about what this layer does not settle (rate, terms, decision) and which chapters settle each. (The Underwriting File; §10.5, §10.6)
  2. Harbor Steel also has "several workers'-comp claims" in the file. Without inventing figures, explain how the credibility of the workers'-comp experience is likely to differ from the credibility of the property fire experience for an account this size — and why that difference matters for which pieces of the program you price more on own-experience and which more on the class. (§10.5; preview of the X-mod, Ch. 22)
  3. The capstone (Chapter 40) will state a bind-with-conditions decision. Explain why nothing in this chapter lets you make that decision yet — what specifically is still missing after the math is done — and name the three later chapters that supply the missing pieces. (The Underwriting File; §10.3, and the process from §7.1)