Chapter 28 Exercises

Work these with the chapter's central habit of mind: behind every account is not just a loss ratio but a claim on the company's finite, expensive surplus — so ask of every risk how much capital does this consume, and does it earn the cost of that capital? 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 (§28.3) point you back to the relevant part of the chapter. Treat every dollar figure as a constructed teaching example.

A. Recall and definitions

  1. † Define policyholder surplus in one sentence, and explain in one more why it is named for policyholders rather than shareholders. (§28.1)
  2. State the difference between loss reserves and surplus: what loss does each pay for? (§28.1)
  3. Write the formula for the premium-to-surplus ratio and explain in one line what a "2-to-1" ratio means. (§28.2)
  4. † Name the four major risk categories the RBC formula charges capital for, and give a one-line reason each matters. (§28.3)
  5. List the five RBC action levels in order from "no action" to "mandatory control," with the approximate ratio threshold for each. (§28.3)
  6. Define Solvency II's Solvency Capital Requirement (SCR) in plain language, including its confidence level and time horizon. (§28.4)
  7. † Define enterprise risk management (ERM) and explain, in one sentence, how it differs from traditional siloed risk management. (§28.5)
  8. Expand the acronym ORSA and say, for each word, what it contributes to the meaning. (§28.5)
  9. Define the cost of capital and explain why it makes some combined-ratio-profitable accounts value-destroying. (§28.6)
  10. What is a rating-agency capital model, and why does it usually bind tighter than the regulatory RBC floor? (§28.7)

B. Leverage and the premium-to-surplus ratio

  1. † A carrier holds \$600M of policyholder surplus and writes \$900M of net written premium. (a) What is its premium-to-surplus ratio? (b) If it misses its planned loss ratio by 6 points, roughly how much surplus does it lose? (c) Answer (b) again assuming the same surplus but \$1,800M of net premium, and explain what changed. (§28.2)
  2. Two carriers each hold \$250M of surplus. Carrier A writes \$250M of net premium; Carrier B writes \$625M. Explain which is more leveraged and why an identical underwriting miss hurts B more. (§28.2)
  3. Explain the difference between premium leverage and reserve leverage, and say which one matters most for a workers'-compensation book and why. (§28.2)
  4. † A carrier reports a comfortable 1.1-to-1 premium-to-surplus ratio and is praised by its board as well-capitalized. Name three distinct things that ratio cannot see, each of which could nonetheless put its surplus in real danger. (§28.2, §28.3)

C. Risk-based capital

  1. Explain in your own words why the RBC formula takes a square root of the sum of squared charges rather than simply adding them. What real-world fact does that step rely on? (§28.3)
  2. † A simplified RBC calculation produces these charges (\$M): asset risk 50, credit risk 30, reserve risk 80, premium risk 60. (a) What is the naive sum? (b) Compute the covariance-adjusted figure as the square root of the sum of the squares. (c) Explain why the adjusted number is so much lower, and name one real scenario in which relying on it would be dangerous. (§28.3)
  3. A carrier reports an RBC ratio (total adjusted capital ÷ authorized control level) of 165%. Which action level is it in, what is it required to do, and why is "165%" less reassuring than it might sound to a layperson? (§28.3)
  4. † Connect Chapter 27 to this chapter: which RBC risk charge penalizes a carrier for reinsurance it may not be able to collect, and why does that mean an underwriter's choice of reinsurer is also a capital decision? (§28.3; Ch. 27)
  5. Explain what RBC catches well and what it misses, using a coastal-concentrated carrier as your example of the miss. (§28.3)
  6. Why does the NAIC model law forbid using RBC results in advertising or to rank insurers? Tie your answer to the book's adverse-selection / death-spiral theme. (§28.3; Ch. 1)

D. Solvency II, ERM, and the ORSA

  1. † Build a comparison: list three concrete ways Solvency II differs from U.S. RBC (approach, calibration, internal models, valuation basis, disclosure — pick three) and say which difference matters most for pricing a catastrophe-exposed account, and why. (§28.4)
  2. Explain why a "1-in-200-year" capital standard means a catastrophe account's capital charge comes from "the monstrous hurricane, not the average season." (§28.4)
  3. Give a concrete example of an aggregate risk that no single underwriter, actuary, or investment officer would see in isolation but that ERM exists to govern. (§28.5)
  4. † Explain what a catastrophe stress test inside an ORSA does, and why the aggregate of a company's coastal accounts — Harbor Steel among them — is exactly what it sums up. Tie this to Chapter 1's independence assumption. (§28.5; Ch. 1)

E. The cost of capital and return on capital

  1. † Account X runs a 92% combined ratio (8% underwriting profit) and ties up capital that costs the company 5% of premium. Account Y runs a 98% combined ratio (2% profit) and ties up capital costing 1% of premium. (a) Compute each account's economic profit (underwriting profit − capital cost) as a percentage of premium. (b) Which account does the company prefer, and what does this show about judging accounts on combined ratio alone? (§28.6)
  2. Explain, in your own words, why a catastrophe-exposed account must be priced to a lower combined ratio than an economically equivalent inland account just to create the same value for the company. (§28.6)
  3. † Define RAROC (risk-adjusted return on capital) and explain how it lets a company compare a coastal property line with an inland casualty line on the same footing despite their very different capital appetites. (§28.6)
  4. The chapter calls the cost-of-capital trap "the most sophisticated in the book." Restate the trap in two sentences, and say what the disciplined underwriter does instead. (§28.6)

F. Rating-agency capital

  1. Explain why, for most carriers, the rating-agency capital model — not the regulatory RBC floor — is the constraint that actually governs how much catastrophe business an underwriter can write. (§28.7)
  2. † Describe the "stack" of capital requirements (regulatory RBC floor, Solvency II SCR, the company's own ERM buffer, the rating-agency requirement) and explain which one a carrier manages to and why a rating downgrade can act like a slow death spiral. (§28.7; Ch. 3)

G. Applied: underwrite, price, and the red flag

  1. † Underwrite this submission. A broker brings you a large, attractively-priced book of coastal condominium property. The loss experience over the last five (calm) years is excellent and the indicated combined ratio is 94%. Your CUO says the company "doesn't have the capital for it." Explain, using this chapter, what the CUO means even though the account is "profitable," what question you would ask before responding, and what you would need to be true to write any of it. (§28.6, §28.7)
  2. Price this risk. An inland account and a coastal account are each offered at \$100 of premium and each is expected to run a 95% combined ratio. The inland account ties up \$40 of capital; the coastal account ties up \$120. Your company's cost of capital is 10%. (a) Compute each account's economic profit. (b) What combined ratio would the coastal account have to be priced to in order to match the inland account's economic profit? (Show the target underwriting profit in dollars, then convert to a combined ratio.) (§28.6)
  3. † Find the red flag. A fast-growing carrier reports record premium growth, a 1.4-to-1 premium-to-surplus ratio it calls "well within tolerance," an RBC ratio of 210%, and a book heavily weighted to a single hurricane-exposed state. Identify the red flags a capital-literate underwriter sees that the headline numbers hide, and explain which single event could turn this picture catastrophic. (§28.2, §28.3, §28.5)
  4. Find the red flag. A long-tail casualty carrier shows a strong premium-to-surplus ratio and a high RBC ratio, but its reserves have "developed adversely" three years running. Why might this carrier's surplus be more endangered than its headline ratios suggest? (§28.2)

H. Memo, ethics, and the Underwriting File

  1. † Write the memo. In one short paragraph to your underwriting committee, justify charging Harbor Steel a margin that "looks rich" relative to its loss experience. Use the cost-of-capital argument explicitly, and name what you are protecting. (§28.6, The Underwriting File)
  2. Ethics dilemma. Your company is near the rating-agency capital limit for coastal exposure. A long-standing coastal community — homes and small businesses that genuinely need coverage — is losing access as carriers pull back. Writing them would strain the rating; declining them deepens a protection gap. Lay out the genuine tension between the company's solvency duty and insurance's social function, and explain why "just write it, they need it" is not an answer a solvent insurer can give. (§28.1, §28.6; Ch. 1)
  3. Ethics / governance. The RBC model law keeps a near-action-level carrier's results confidential and bars using RBC to rank insurers. Argue both sides: how does this confidentiality protect policyholders, and what is the cost of a public that cannot easily tell a strong insurer from a weak one? (§28.3)
  4. † Underwriting File extension. Chapter 27 ceded Harbor Steel's catastrophe exposure to the cat XOL treaty over a roughly \$5M net retention. Explain how that reinsurance decision changes the capital this chapter assigns to the account — both reducing it (the cat exposure is largely ceded) and creating a new, smaller charge (the reinsurance recoverable). State the running disposition this chapter reaches and the two questions it explicitly leaves to Chapters 29 and 30. (The Underwriting File; Ch. 27)
  5. Underwriting File extension. Suppose the broker succeeds in negotiating Harbor Steel's price down to a "merely adequate" combined ratio that still clears your loss-ratio hurdle. Using §28.6, explain precisely why binding at that price would be a mistake even if no hurricane ever arrives. (§28.6, The Underwriting File)
  6. Synthesis. In a few sentences, reconcile two claims this book has made: Chapter 3 said "the combined ratio tells the truth," and this chapter said a 95% combined ratio can destroy company value. Are they contradictory? Explain what each truth is about, and how a senior underwriter holds both at once. (§28.6; Ch. 3)