Chapter 10 Quiz

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Chapter 10 Quiz

Twenty questions to check your grasp of the mathematics of risk and price. Answers and brief explanations are in the collapsed key at the bottom — try the whole quiz before you open it. All figures are constructed teaching numbers.

Part 1 — Multiple choice (15)

1. Expected loss for a period is best decomposed as:

A. paid losses minus reserves
B. frequency × severity
C. written premium × loss ratio
D. earned premium ÷ exposures

2. A severity distribution is typically:

A. symmetric around its mean
B. low-count and lumpy
C. skewed with a long right tail, so the mean exceeds the median
D. uniform across all loss sizes

3. The honest form of the loss ratio is:

A. paid losses ÷ written premium
B. incurred losses ÷ earned premium
C. paid losses ÷ earned premium
D. incurred losses ÷ written premium

4. "Incurred losses" equal:

A. paid losses only
B. paid losses plus case reserves plus IBNR
C. case reserves only
D. written premium minus expenses

5. A fast-growing, long-tail book that reports a beautiful loss ratio in year one is most likely flattered because:

A. its expenses are unusually low
B. it is using earned premium and incurred losses
C. fresh written premium has not yet earned or produced its losses, and the losses have not developed
D. the law of large numbers has stabilized it

6. The pure premium (loss cost) is:

A. the premium after all expense and profit loads
B. the expected loss per unit of exposure
C. the deductible plus the limit
D. the loss ratio times the expense ratio

7. A good exposure base should:

A. be easy to negotiate down at audit
B. stay constant regardless of the size of the risk
C. vary with the expected loss and be hard to manipulate
D. always be the number of employees

8. Development corrects for:

A. inflation in rebuild costs between years
B. the immaturity of open and not-yet-reported claims as they grow toward their ultimate value
C. the insurer's expense ratio
D. changes in the exposure base

9. Trend corrects for:

A. open claims settling above their reserves
B. IBNR
C. change over time in the cost and frequency of losses between the data period and the period being priced
D. the credibility of the sample

10. In classical (limited-fluctuation) credibility, the square-root rule says $Z$ equals:

A. $n / N$
B. $\sqrt{n / N}$
C. $n^2 / N$
D. $N / n$

11. Against a full-credibility standard of 1,000 claims, an account with 10 claims has a credibility of about:

A. 1.00
B. 0.50
C. 0.10
D. 0.01

12. The credibility-weighted estimate is:

A. $Z \times \text{class} + (1-Z) \times \text{own}$
B. $Z \times \text{own} + (1-Z) \times \text{class}$
C. $\text{own} + \text{class}$
D. $Z \times (\text{own} - \text{class})$

13. An account's own loss ratio is 100%, the class is 60%, and $Z = 0.25$. The credibility-weighted loss ratio is:

A. 60%
B. 70%
C. 80%
D. 100%

14. In Bühlmann credibility, $Z = n/(n+K)$ with $K = \text{EPV} / \text{VHM}$. Credibility is high when:

A. between-risk differences are small and within-risk noise is large
B. between-risk differences are large and within-risk noise is small
C. both variances are equal
D. $n$ is small

15. A life-insurance applicant's own-experience credibility for the event being priced (their death) is essentially:

A. full (1.0), because each person is unique
B. zero, so the underwriter relies on classification against mortality tables
C. 0.50, by convention
D. set by the applicant's age alone, with no class involved

Part 2 — Short answer (5)

16. Explain why an underwriter keeps frequency and severity separate when reading a loss run instead of working from total-losses-divided-by-years. Name one control that addresses a frequency problem and one that addresses a severity problem.

17. Define the permissible (target) loss ratio and show how it is computed from the expense and profit loads. Why is it the bridge between this chapter and pricing?

18. A claim from three years ago is valued today at \$200,000 and is still open. Describe — in words, no need to compute exactly — the two adjustments you would make before pricing off it, and which direction each moves the number.

19. Two fires in five years on a commercial property account: explain why credibility theory says not to over-react, and then name the single circumstance under which an underwriter should deliberately price above the credibility-weighted number.

20. State the Bühlmann question in plain English and explain why a catastrophe-exposed coastal property account's clean five-year record should not be given high own-experience credibility.

Answer key (try the quiz first)

**Multiple choice:** 1. **B** — expected loss = frequency × severity (§10.1). 2. **C** — severity is right-skewed; the mean sits above the median, pulled up by the tail (§10.1). 3. **B** — incurred losses ÷ earned premium is the truth-telling form (§10.2). 4. **B** — incurred = paid + case reserves + IBNR (§10.2). 5. **C** — written premium that has not earned, and losses that have not arrived or developed, flatter the ratio (§10.2, §10.4). 6. **B** — the pure premium / loss cost is expected loss per exposure unit, before loads (§10.3). 7. **C** — a good base varies with expected loss and resists manipulation (§10.3). 8. **B** — development addresses immaturity (open claims growing, IBNR) (§10.4). 9. **C** — trend addresses change over time in loss cost and frequency (§10.4). 10. **B** — $Z = \sqrt{n/N}$ (§10.5). 11. **C** — $\sqrt{10/1000} = \sqrt{0.01} = 0.10$ (§10.5). 12. **B** — $Z \times \text{own} + (1-Z) \times \text{class}$ (§10.6). 13. **B** — $0.25 \times 100\% + 0.75 \times 60\% = 25\% + 45\% = 70\%$ (§10.6). 14. **B** — large between-risk variance (VHM) and small within-risk noise (EPV) make $K$ small and $Z$ high (§10.7). 15. **B** — an individual's death is a one-time event with no own experience; the class (mortality tables) does the work, so classification is everything (§10.6, Ch. 17). **Short answer (model points):** 16. Frequency and severity are driven by different hazards and fixed by different tools; a single losses-÷-years figure blends a frequency signal and a severity signal you needed to keep apart, hiding *which* account you are pricing. A higher **deductible** addresses a frequency (many-small-claims) problem; a **policy limit / reinsurance** addresses a severity (rare-but-huge) problem. (§10.1) 17. Permissible loss ratio = 1 − (expense load + profit/contingency load). If expenses are ~30% and profit ~5%, the permissible loss ratio is 65%. It is the bridge to pricing because building a rate (Chapter 11) is really solving for the premium at which the *expected* loss ratio equals the permissible one; writing above it is planned underwriting loss. (§10.2) 18. **Develop** the \$200,000 toward its ultimate value (open claims and IBNR grow it — moves the number *up*), then **trend** the developed figure to the future cost level you are pricing (inflation/social inflation — moves it *up* again). The raw figure on the loss run understates the true future cost. (§10.4) 19. Two claims is a single-digit credibility on any reasonable standard, so the account's own experience earns only a small weight and the *class* should carry most of the price; the two fires are a low-count, severity-skewed sample, mostly noise. The exception: when a *qualitative* read shows the experience reflects a **changed or live hazard** (e.g., a recurring hot-work fire risk) rather than random fluctuation, the underwriter overrides the blend *upward* with documented reasons, because credibility math is blind to causation. (§10.5–10.6) 20. The Bühlmann question: "Is the difference I'm seeing between this risk and the class a real, persistent difference (signal) or the random bounce of a small, skewed sample (noise)?" Catastrophe-exposed coastal property is dominated by *rare, random large losses*, so a clean five-year window is mostly the absence of a rare event, not evidence of a genuinely better risk; the within-risk noise (EPV) is huge, $K$ is large, and own-experience credibility should stay *low*. (§10.7)