A policy has ground-up loss X, ordinary deductible 500, and insurer payment limit 2000. Write the insurer payment variable.

The insurer payment is Y = min((X - 500)_+, 2000). The 2000 limit caps the insurer payment, not the ground-up loss.

N is Poisson with mean 10. Individual claim severity has mean 200 and variance 90,000. Find E[S] and Var(S) for S = X_1 + ... + X_N.

E[S] = 10(200) = 2000. For compound Poisson, Var(S) = lambda E[X^2] = 10(90,000 + 200^2) = 1,300,000. Steps: Use E[S] = E[N]E[X]. Compute E[X^2] = Var(X) + E[X]^2. For Poisson frequency, use Var(S) = lambda E[X^2].

A limited fluctuation credibility rule requires the observed average to be within 5% of the true mean with probability 95%. What does full credibility mean in words?

Full credibility means the exposure or claim count is large enough that the observed average meets that precision and probability standard, so the experience can be used without partial credibility blending under that rule.

Projected losses are 720,000 for 6000 exposures. Fixed expense is 8 per exposure and variable expense plus profit load is 20% of premium. Find the indicated average rate per exposure.

Loss cost is 720,000 / 6000 = 120. Add fixed expense to get 128. Divide by 1 - 0.20, so the indicated rate is 128 / 0.80 = 160. Steps: Compute projected loss cost per exposure. Add fixed expense per exposure. Divide by one minus the variable expense and profit load.

Exam guide

FAM Short-Term Models

FAM short-term models connect insurance payment rules to severity, frequency, aggregate risk, credibility, ratemaking, and reserving calculations.

Credential side

SOA

Primary intent

FAM short-term models

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Loss Distribution Formula Atlas

Quick Answer

The short-term side of FAM asks candidates to translate insurance contract language into loss variables, payment variables, aggregate risk variables, and rate or reserve calculations.

The highest-value habit is to name the object before calculating. Is the question about ground-up loss, insurer payment, reinsurer payment, aggregate loss, expected stop-loss payment, indicated rate, or outstanding claims?

Coverage Modifications

Coverage modifications include deductibles, policy limits, coinsurance, inflation, loss elimination ratios, proportional reinsurance, and excess-of-loss reinsurance. The notation note matters here because FAM uses policy limit for the maximum insurer payment and maximum covered loss for the loss level above which no additional benefit is paid.

For an ordinary deductible, start from the payment variable. For reinsurance, write the insurer and reinsurer shares separately before taking expectations.

Loss elimination ratio

\operatorname{LER}(d)=\frac{E[X\wedge d]}{E[X]}

Payment with deductible d and payment limit u

Y=\min\{(X-d)_{+},u\},\qquad E[Y]=E[X\wedge(d+u)]-E[X\wedge d]

Severity And Frequency

Severity models describe claim amounts. Frequency models describe claim counts. FAM expects candidates to recognize parameter roles, moment existence, distribution relationships, and whether a count model belongs to the recursive classes used in aggregate calculations.

The practical split is simple: use severity formulas for claim size, frequency formulas for claim count, and aggregate formulas once the portfolio total is the object.

Collective risk model

S=X_1+\cdots+X_N

Aggregate mean and variance

E[S]=E[N]E[X],\qquad \operatorname{Var}(S)=E[N]\operatorname{Var}(X)+\operatorname{Var}(N)E[X]^2

Aggregate Risk

FAM aggregate questions can ask for mean, variance, convolution, normal or lognormal approximation, stop-loss expected payment, VaR, or Tail VaR. The computational method should follow the question type.

If the severity is discrete and the count family is recursive, Panjer-style thinking belongs nearby. If the question only asks for mean and variance, do not overbuild the distribution.

Estimation And Credibility

FAM introduces maximum likelihood estimation for complete, grouped, truncated, and censored data. It also introduces limited fluctuation credibility. Those are not disconnected topics: both are ways to decide how much trust to place in observed data.

For limited fluctuation credibility, the calculation is only half the answer. The interpretation is whether observed experience is large enough to be treated as fully credible under the stated precision and probability standard.

Limited fluctuation full credibility shape

P\left(\left|\bar X-\mu\right|\le k\mu\right)\ge p

Pricing And Reserving

The short-term pricing and reserving block covers expected loss ratio, chain-ladder, Bornhuetter-Ferguson, ratemaking data adjustments, development, trend, premium at current rate level, expenses, profit loading, loss cost rates, and loss ratio rates.

Reserve questions ask what remains unpaid. Ratemaking questions ask what rate should be charged for future exposure. Both use loss data, but the valuation date and target quantity are different.

Practice

Original Source-Backed Practice

4 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.