Risk theory and loss distributions

Actuarial Science (B.Sc) · General insurance and risk · free study notes for exams, placements and interviews

In short

Risk theory models the total claim amount an insurer will pay over a period as a random sum: a random number of claims, each of a random size. Loss distribution theory supplies the severity models (exponential, gamma, lognormal, Pareto, Weibull, Burr) and frequency models (Poisson, binomial, negative binomial) that feed it, and the mathematics of how deductibles, policy limits and reinsurance reshape them. It is the core of IAI's CS2/CM2-type papers and of university papers titled "General insurance and risk", and it is tested in placement rounds at Swiss Re, Munich Re, RGA, Milliman, WTW, Aon, Mercer, Deloitte, EY, SBI General, ICICI Lombard, Bajaj Allianz, GIC Re and Tata AIG. In practice it drives reinsurance structuring, capital/solvency modelling under the IRDAI regime, reserve volatility estimates and catastrophe pricing.

Explained

Individual versus collective risk models

Individual risk model: S = Y₁ + Y₂ + … + Y_n over a *fixed* number n of policies; each Y_j = 0 with probability (1−q_j) and a positive loss otherwise. Used for group life and for short-term portfolios with known exposure.
Collective risk model: S = X₁ + X₂ + … + X_N, where N is the *random* claim count and the X_i are i.i.d. claim sizes, independent of N. Preferred in general insurance because claim counts are genuinely random.
Both give the same aggregate concept; the collective model is far easier to work with analytically because of the compound-distribution structure.
Key assumptions: claim sizes i.i.d.; independent of frequency; each claim settled in the period. Reality breaks all three (inflation trends, dependence in catastrophes, reporting delay → IBNR), so the model is a base case to be stress-tested.

Frequency distributions

Poisson(λ): mean = variance = λ. Natural for rare independent events; closed under addition and under thinning (if each claim is "large" with probability p, large claims are Poisson(λp)).
Binomial(n, q): variance < mean; a fixed number of risks each of which can give at most one claim.
Negative binomial(k, p): variance > mean — over-dispersion, the usual empirical finding in motor and health. Arises as a mixed Poisson where λ itself is Gamma distributed (heterogeneous drivers).
The (a, b, 0) class covers all three: p_k/p_{k−1} = a + b/k. Poisson a = 0; binomial a < 0; negative binomial 0 < a < 1. The (a, b, 1) class modifies or truncates the probability at zero (zero-inflated motor claims).

Severity (loss) distributions

Exponential(λ): memoryless, mean 1/λ; the simplest severity, mathematically tractable, light-tailed.
Gamma(α, λ): sum of exponentials; moderate tail; common for attritional claims.
Lognormal(μ, σ²): log of the loss is normal; positively skewed, heavier than gamma; the workhorse for fire, liability and motor own-damage severity. Fit by regressing/averaging log-losses.
Pareto (two-parameter, α, λ): F(x) = 1 − (λ/(λ+x))^α; heavy-tailed; k-th moment exists only if α > k. The standard model for large/liability/catastrophe losses. A single-parameter Pareto on x > λ is used above a threshold.
Weibull(c, γ): γ < 1 heavy tail, γ = 1 exponential, γ > 1 light tail; used for reinsurance layers.
Burr: Pareto generalisation with an extra shape parameter — flexible tail.
Tail classification: compare limiting behaviour of survival functions, or use the mean residual life e(x) = E[X − x | X > x]: increasing in x for Pareto (heavy), constant for exponential, decreasing for light tails. The coefficient of skewness and the ratio of high percentiles to the mean are exam-standard discriminators.
Estimation: method of moments (equate sample and theoretical moments), maximum likelihood (preferred — asymptotically efficient), percentile matching. Fit checked by Q-Q plot, chi-square and Kolmogorov-Smirnov tests, and by comparing observed and fitted excess layers.

Aggregate claims: moments, MGF and recursion

For the compound model, condition on N: E[S] = E[N]·E[X]; Var(S) = E[N]·Var(X) + Var(N)·(E[X])².
Compound Poisson: Var(S) = λ·E[X²], and the sum of independent compound Poisson risks is again compound Poisson with λ = Σλ_i and severity a λ-weighted mixture — the property that lets portfolios be combined.
MGF relation: M_S(t) = M_N(ln M_X(t)) = P_N(M_X(t)) where P_N is the probability generating function. Differentiating gives the moments; the third central moment gives skewness, always positive for compound Poisson with positive claims.
Panjer recursion: for an (a, b, 0) frequency and a *discretised* severity on 0, 1, 2, … multiples of a span h, the exact aggregate distribution is built up recursively — the standard computational method before simulation.
Approximations: normal (adequate only for large λ and light tails; ignores skew), translated gamma and normal power (NP2) which match three moments and so capture skewness far better in the right tail.
Monte Carlo simulation is what is actually used in practice: draw N, then N severities, repeat 10⁵–10⁶ times, read percentiles for the 99.5% capital figure.

Reinsurance, deductibles and limits

Excess of loss (XL) with retention M: insurer pays Y = min(X, M), reinsurer Z = max(0, X − M). E[Z] = ∫_M^∞ (x − M) f(x) dx = ∫_M^∞ S(x) dx.
Proportional (quota share) α: insurer αX, reinsurer (1−α)X — leaves the shape of the distribution unchanged, only scales it; cheap and simple but gives no protection against a single huge loss.
Policy deductible d (ordinary): insurer pays max(0, X − d). Two different quantities are examined — payment per loss (zero when X ≤ d) and payment per payment (conditional on X > d, the loss elimination is already removed); confusing them is the single most common error.
Loss elimination ratio (LER) = E[min(X, d)]/E[X] — the fraction of expected cost removed by a deductible.
Franchise deductible: nothing paid if X ≤ d, but the full X paid if X > d.
Policy limit u / coverage cap: payment min(X, u). A layer (d, u) pays E[min(X,u)] − E[min(X,d)].
Reinsurer's claim count: with Poisson(λ) gross frequency, claims exceeding M arrive as Poisson(λ·S(M)) — thinning. For Pareto and single-parameter Pareto severities the *conditional* excess is again Pareto, which is why Pareto dominates XL pricing.
Inflation: if losses inflate by factor (1+i) but the retention M is fixed, the reinsurer's share rises faster than inflation — leveraged effect of inflation on excess layers, a favourite exam question.
Stop-loss reinsurance applies to the aggregate S: reinsurer pays max(0, S − k); E[(S−k)₊] = Σ_{s>k}(s−k)P(S=s) for discrete S.

Ruin theory and premium principles

Surplus process: U(t) = u + ct − S(t), with initial surplus u, premium income rate c, aggregate claims S(t) (compound Poisson in the classical Cramér-Lundberg model).
Relative security loading θ: c = (1+θ)λ·E[X]; ruin is certain if θ ≤ 0 (net profit condition c > λE[X]).
Adjustment coefficient R: the positive root of λ·M_X(R) = λ + cR, equivalently λ + cR − λM_X(R) = 0. Larger R means a safer portfolio.
Lundberg's inequality: ψ(u) ≤ e^{−Ru} — ruin probability decays exponentially in initial surplus. For exponential claims with mean 1/λ_X the exact result is ψ(u) = (1/(1+θ))·e^{−Ru} with R = θλ_X/(1+θ).
Effect of parameters: ψ(u) decreases with u and θ; increases with claim-size variability; reinsurance lowers both the mean and the variance of retained claims, usually raising R even though net premium income falls — the basis of "optimal retention" questions.
Premium principles: pure/expected value (1+θ)E[S]; variance principle E[S] + βVar(S); standard deviation principle E[S] + β·√Var(S); exponential principle (1/β)ln M_S(β); percentile principle (premium = the p-th percentile of S). Desirable properties tested: non-negative loading, additivity, consistency, scale invariance, no rip-off.
Risk measures used for capital: VaR_p(S) = the p-th quantile; TVaR/CTE = E[S | S > VaR_p] — coherent (sub-additive), unlike VaR, and preferred for tail risk.

Extreme values, data issues and Indian practice

Extreme value theory: block maxima follow a GEV distribution (Gumbel, Fréchet, Weibull types); exceedances above a high threshold follow the generalised Pareto distribution (GPD) — the peaks-over-threshold method used for cyclone, flood and earthquake layers.
Data adjustments before fitting: trend/inflate historical losses to current values; allow for changes in exposure, deductibles and limits; handle censoring (policy limits) and truncation (deductibles) correctly in the likelihood; strip out large one-off events and model them separately.
IBNR and IBNER: incurred but not reported / not enough reported — aggregate claims for an accident year are not fully known, so chain-ladder and Bornhuetter-Ferguson estimates feed the severity data (reserving itself is a separate topic).
Indian regulatory setting: general insurers are regulated by IRDAI under the Insurance Act, 1938 and the IRDAI Act, 1999. Solvency is a required-solvency-margin / available-solvency-margin ratio, with the statutory minimum control level set at 1.5 times (150%); a risk-based capital framework has been under development and piloting rather than fully enforced, so quote the factor-based RSM regime as current. Reinsurance cessions follow the order of preference in the IRDAI reinsurance regulations, with Indian reinsurers (GIC Re) having first right of refusal — the precise regulation and obligatory cession percentage have been revised several times, so cite the current regulation rather than a fixed number.
Motor third-party rates are notified by the Ministry of Road Transport and Highways in consultation with IRDAI (IRDAI itself notified them up to FY 2021-22); own-damage and most other lines are file-and-use, so severity modelling directly drives filed rates.

Key points

Collective risk model - S = ΣX_i over random N; the standard general-insurance aggregate model.
Individual risk model - fixed number of policies, each contributing 0 or a loss; used for group schemes.
E[S] = E[N]E[X] - aggregate mean is a simple product; variance is not.
Var(S) = E[N]Var(X) + Var(N)E[X]² - both frequency and severity uncertainty contribute.
Compound Poisson - Var(S) = λE[X²]; sums of independent compound Poissons stay compound Poisson.
Over-dispersion - variance > mean in claim counts; use negative binomial (a Gamma-mixed Poisson).
(a, b, 0) class - Poisson, binomial, negative binomial unified by p_k/p_{k−1} = a + b/k; prerequisite for Panjer.
Panjer recursion - exact aggregate distribution from a discretised severity, no simulation needed.
Heavy tail - Pareto/Burr/Weibull(γ<1); Pareto's k-th moment exists only if α > k.
Mean residual life - increasing ⇒ heavy tail; constant ⇒ exponential; the clean tail test.
Excess of loss - reinsurer pays max(0, X − M); E[Z] = ∫_M^∞ S(x)dx.
Thinning - with Poisson frequency, claims above M are Poisson(λS(M)).
Leveraged inflation - a fixed retention makes the reinsurer's cost grow faster than inflation.
LER - E[min(X,d)]/E[X]; the share of expected loss a deductible removes.
Adjustment coefficient R - root of λM_X(R) = λ + cR; bigger R, safer portfolio.
Lundberg's inequality - ψ(u) ≤ e^{−Ru}; ruin risk falls exponentially with surplus.
VaR vs TVaR - VaR is a quantile and not sub-additive; TVaR averages the tail and is coherent.
IRDAI solvency - factor-based required solvency margin with a 150% minimum control level.

Formulas and examples

Individual risk model: S = Σ_{j=1}^{n} Y_j, n fixed; E[S] = Σ E[Y_j], Var(S) = Σ Var(Y_j) (independent policies)
Collective risk model: S = Σ_{i=1}^{N} X_i, N random, X_i i.i.d., independent of N
E[S] = E[N]·E[X]; Var(S) = E[N]·Var(X) + Var(N)·(E[X])² (₹ and ₹² respectively)
Compound Poisson(λ, X): E[S] = λE[X], Var(S) = λE[X²], E[(S−E[S])³] = λE[X³]
Skewness(S) = λE[X³] / (λE[X²])^{3/2} (dimensionless)
MGF: M_S(t) = P_N(M_X(t)) = M_N(ln M_X(t)); P_N(z) = E[z^N]
Panjer recursion (a,b,0), f_j = P(X = j), p₀ = P(N=0): g₀ = P_N(f₀); g_x = [1/(1 − a f₀)] · Σ_{j=1}^{x} (a + bj/x) f_j g_{x−j}
Pareto(α, λ): S(x) = (λ/(λ+x))^α; E[X] = λ/(α−1), α>1; E[X²] = 2λ²/((α−1)(α−2)), α>2
Lognormal(μ, σ²): E[X] = e^{μ+σ²/2}; E[X²] = e^{2μ+2σ²}; Var = e^{2μ+σ²}(e^{σ²} − 1)
Exponential(λ): E[X] = 1/λ, E[X²] = 2/λ², E[Xᵏ] = k!/λᵏ
Gamma(α, λ): E[X] = α/λ, Var = α/λ²
Deductible d, per loss: E[(X−d)₊] = ∫_d^∞ (x−d) f(x) dx = ∫_d^∞ S(x) dx
Per payment (conditional): E[X−d | X>d] = E[(X−d)₊] / S(d)
Limited expected value: E[min(X, u)] = ∫_0^u S(x) dx; layer (d,u) cost = E[min(X,u)] − E[min(X,d)]
Loss elimination ratio LER = E[min(X,d)] / E[X] (dimensionless)
XL retention M: insurer E[min(X,M)], reinsurer E[(X−M)₊]; reinsurer frequency λ_R = λ·S(M)
Surplus: U(t) = u + ct − S(t); c = (1+θ)·λ·E[X] (₹ per unit time)
Adjustment coefficient: λ + cR = λ·M_X(R); Lundberg: ψ(u) ≤ e^{−Ru}
Exponential claims mean μ: R = θ/(μ(1+θ)); ψ(u) = (1/(1+θ))·e^{−Ru}
VaR_p(S) = inf{s : F_S(s) ≥ p}; TVaR_p(S) = E[S | S > VaR_p(S)]
Premium principles: expected value (1+θ)E[S]; variance E[S]+βVar(S); std-dev E[S]+β√Var(S); exponential (1/β)ln M_S(β)

Symbols: λ = Poisson claim rate per period (claims), X = individual claim size (₹), N = claim count, S = aggregate claims (₹), M = retention (₹), d = deductible (₹), u = policy limit (₹), θ = premium loading (dimensionless), c = premium income rate (₹/year), u in U(t) = initial surplus (₹), R = adjustment coefficient (per ₹).

Example 1 — Aggregate mean, variance and skewness (compound Poisson). A motor own-damage portfolio has Poisson claim frequency λ = 500 per year. Claim sizes are exponential with mean ₹40,000.

E[X] = 40,000; E[X²] = 2/λ_X² = 2 × 40,000² = 3.2 × 10⁹; E[X³] = 6 × 40,000³ = 3.84 × 10¹⁴.
E[S] = 500 × 40,000 = ₹2,00,00,000 (₹2 crore).
Var(S) = λE[X²] = 500 × 3.2 × 10⁹ = 1.6 × 10¹² → SD = ₹12,64,911.
Skewness = λE[X³]/(λE[X²])^{1.5} = (500 × 3.84×10¹⁴)/(1.6×10¹²)^{1.5}. Numerator = 1.92 × 10¹⁷; denominator = (1.6×10¹²)^{1.5} = 1.6^{1.5} × 10¹⁸ = 2.0239 × 10¹⁸ → skewness ≈ 0.0949 (mildly right-skewed, so a normal approximation is defensible here).
Premium at θ = 20%: c = 1.2 × 2 crore = ₹2.4 crore; at the 99.5% normal approximation, capital ≈ E[S] + 2.576 SD = 2,00,00,000 + 32,58,412 ≈ ₹2.33 crore, i.e. about ₹33 lakh above the mean.

Example 2 — Negative binomial frequency changes the answer. Same severity (exponential, mean ₹40,000, Var = 1.6 × 10⁹), but N is negative binomial with E[N] = 500 and Var(N) = 1,500 (over-dispersion factor 3).

E[S] = 500 × 40,000 = ₹2 crore (unchanged).
Var(S) = 500 × 1.6×10⁹ + 1,500 × (40,000)² = 8.0 × 10¹¹ + 2.4 × 10¹² = 3.2 × 10¹² → SD = ₹17,88,854.
SD is 1.414 times the Poisson case, so the 99.5% capital add-on rises from ₹32.6 lakh to 2.576 × 17,88,854 ≈ ₹46.1 lakh. Ignoring over-dispersion understates required capital by about 29%.

Example 3 — Excess of loss with Pareto severity, and leveraged inflation. Claim sizes are Pareto with α = 3, λ = ₹2,00,000. Retention M = ₹5,00,000. Gross frequency Poisson 400 per year.

E[X] = λ/(α−1) = 2,00,000/2 = ₹1,00,000; gross expected claims = 400 × 1,00,000 = ₹4 crore.
S(M) = (2,00,000/7,00,000)³ = (2/7)³ = 8/343 = 0.023324 → reinsurer's frequency = 400 × 0.023324 = 9.33 claims/year.
E[(X−M)₊] = ∫_M^∞ S(x)dx = λ^α (λ+M)^{1−α}/(α−1) = [(λ+M)/(α−1)]·S(M) = (7,00,000/2) × 0.023324 = ₹8,163 per gross claim.
Reinsurer's expected annual cost = 400 × 8,163 = ₹32.65 lakh, i.e. 8.16% of the gross ₹4 crore.
Now inflate all claims by 10% (X′ = 1.1X ⇒ Pareto with λ′ = 2,20,000, same α) with M unchanged. S′(M) = (2,20,000/7,20,000)³ = (0.305556)³ = 0.028528; E[(X′−M)₊] = (7,20,000/2) × 0.028528 = ₹10,270. Reinsurer's cost = 400 × 10,270 = ₹41.08 lakh — a rise of 25.8% from a 10% inflation. That is the leveraged effect; the insurer's retained cost rises only from ₹3.673 crore to ₹3.989 crore, i.e. 8.6%.

Example 4 — Deductible, LER and per-payment severity. Claim sizes are exponential with mean ₹50,000 (λ_X = 1/50,000). A deductible d = ₹10,000 is introduced.

Per loss: E[(X−d)₊] = e^{−d/50,000}/λ_X = 50,000 × e^{−0.2} = 50,000 × 0.818731 = ₹40,937.
E[min(X,d)] = E[X] − E[(X−d)₊] = 50,000 − 40,937 = ₹9,063 → LER = 9,063/50,000 = 18.1% of expected cost removed.
Per payment: E[X − d | X > d] = 40,937/e^{−0.2} = 40,937/0.818731 = ₹50,000 — unchanged, because the exponential is memoryless. A pure risk premium per *policy* falls 18.1%, but the average *cheque written* is the same size.
Claim frequency also falls: if gross frequency was Poisson 0.15 per policy-year, the frequency of payments is 0.15 × e^{−0.2} = 0.1228, and the risk premium per policy = 0.1228 × 50,000 = ₹6,141 against 0.15 × 50,000 = ₹7,500 gross — the same 18.1% saving, computed two ways.

Example 5 — Ruin probability and optimal retention. Claims are exponential with mean ₹20,000, Poisson λ = 100 per year, loading θ = 25%, initial surplus u = ₹20 lakh.

c = 1.25 × 100 × 20,000 = ₹25 lakh per year.
R = θ/(μ(1+θ)) = 0.25/(20,000 × 1.25) = 0.25/25,000 = 1 × 10⁻⁵ per ₹.
ψ(u) = (1/1.25)·e^{−1e−5 × 20,00,000} = 0.8 × e^{−20} = 0.8 × 2.061×10⁻⁹ = 1.65 × 10⁻⁹ — negligible.
With surplus of only ₹2 lakh: ψ = 0.8 × e^{−2} = 0.8 × 0.13534 = 0.1083, about an 11% chance of ruin. Surplus matters exponentially.
Lundberg's bound at u = ₹2 lakh is e^{−2} = 0.1353 — an upper bound, correctly above the exact 0.1083.

Severity model	Tail	Moments exist	Typical Indian use
Exponential	Light	All	Teaching, simple attritional claims
Gamma	Light-moderate	All	Motor OD, health attritional
Lognormal	Moderate-heavy	All	Fire, marine, motor OD severity
Weibull (γ<1)	Heavy	All	Reinsurance layers
Pareto (α)	Heavy	Only k < α	Liability, large fire, catastrophe
GPD (ξ>0)	Heavy	Only k < 1/ξ	Cyclone/flood peaks-over-threshold

Reinsurance type	Insurer pays	Reinsurer pays	Protects against
Quota share α	αX	(1−α)X	Overall volume; not a single large loss
Excess of loss M	min(X, M)	max(0, X−M)	Individual large claims
Stop loss k (aggregate)	min(S, k)	max(0, S−k)	Bad accumulation year
Franchise d (policy)	0 if X≤d, else X paid in full	—	Nuisance claims only

Where it is used

General insurance pricing teams (ICICI Lombard, Bajaj Allianz, HDFC ERGO, Tata AIG, SBI General): fitting lognormal/Pareto severity and negative binomial frequency by vehicle class, age and geography for file-and-use rate filings.
Reinsurance and broking (GIC Re, Swiss Re, Munich Re, Aon, Gallagher Re, WTW): pricing XL layers, choosing retentions, exposure and experience rating of layers using Pareto curves.
Capital and solvency work: simulating aggregate losses to a 99.5% percentile and using TVaR for internal capital, alongside IRDAI's required solvency margin reporting and the appointed actuary's certificate.
Catastrophe modelling for Indian cyclone, flood and earthquake exposure — GPD/GEV tails feeding PML (probable maximum loss) estimates used by reinsurers and the Indian Market Terrorism Risk Insurance Pool.
Health and crop insurance: Ayushman Bharat/PMJAY-adjacent schemes and PMFBY crop insurance use frequency-severity models with large aggregate stop-loss style arrangements between insurers and governments.
Roles: actuarial analyst (pricing/reserving/capital), cat modeller, risk analyst in banks' model validation, consulting actuary in Milliman/Mercer/Deloitte/EY.

Interview questions

Q: What is the difference between the individual and collective risk models? A: The individual model sums losses over a fixed number of policies, most contributing zero; the collective model sums a random number N of i.i.d. claim amounts. The collective model is more tractable and standard in general insurance.
Q: For a compound Poisson, why is Var(S) = λE[X²] rather than λVar(X)? A: Substituting E[N] = Var(N) = λ into Var(S) = E[N]Var(X) + Var(N)E[X]² gives λ(Var(X) + E[X]²) = λE[X²].
Q: Claim counts in motor show variance well above the mean. What do you fit and why? A: A negative binomial, because it is a Gamma-mixed Poisson and so models heterogeneity between policyholders; the Poisson forces variance = mean and would understate volatility and capital.
Q: How do you tell whether a severity distribution is heavy-tailed? A: Check whether the mean residual life E[X−x|X>x] increases with x, whether higher moments exist, and how the survival function decays — polynomial decay (Pareto) is heavy, exponential decay is not.
Q: Why is Pareto convenient for excess-of-loss pricing? A: The conditional distribution of the excess above a threshold is again Pareto, so layer costs have closed forms and extrapolation above the observed data range is consistent.
Q: What is the leveraged effect of inflation on an excess layer? A: With a fixed retention, claim inflation pushes proportionally more of each loss above the retention, so the reinsurer's expected cost grows by more than the inflation rate — in the worked case, 25.8% for 10% inflation.
Q: Does a deductible reduce the average size of the payments made? A: Not necessarily. For exponential (memoryless) severity the mean payment per payment is unchanged; the saving comes from fewer payments. For heavy-tailed severity the mean per payment actually rises.
Q: State and interpret Lundberg's inequality. A: ψ(u) ≤ e^{−Ru}: ruin probability falls exponentially in initial surplus, with rate given by the adjustment coefficient R, which is the positive root of λ + cR = λM_X(R).
Q: How does buying reinsurance affect the adjustment coefficient? A: It cuts the retained claim size distribution and its variance, which usually raises R even though the net premium rate falls — up to the retention where the reinsurance loading outweighs the risk reduction.
Q: Why is TVaR preferred to VaR for capital? A: VaR is a single quantile that ignores how bad losses beyond it are and is not sub-additive, so it can penalise diversification; TVaR averages the tail and is a coherent risk measure.
Q: What is the Panjer recursion used for, and what must you do first? A: It computes the exact aggregate claims distribution for (a,b,0) frequencies; the severity must first be discretised onto a lattice (method of rounding or local moment matching).
Q: What solvency requirement does IRDAI impose on a general insurer? A: A factor-based required solvency margin with available solvency margin at least 1.5 times the required level (150% control level); a risk-based capital approach has been under development rather than in force, so the factor basis is what is reported.
Q: Why might a normal approximation to S be dangerous? A: Aggregate claims are positively skewed, so the normal understates the right tail; translated gamma, NP2 or simulation should be used for high percentiles.

Common mistakes

Using Var(S) = λVar(X) for a compound Poisson instead of λE[X²] — a large and common numerical error.
Confusing per loss and per payment expected payments under a deductible: dividing (or not dividing) by S(d).
Quoting a Pareto mean or variance without checking the moment condition α > 1 and α > 2 — with α = 1.5 the variance is infinite, so a variance-based premium principle is meaningless.
Applying the Poisson thinning result λ·S(M) to a negative binomial frequency; only the Poisson keeps its form with the parameter simply scaled — a negative binomial thins to a negative binomial with a changed p, so its mean and variance do not scale the same way.
Forgetting to inflate the retention when projecting a reinsurance programme forward, then being surprised by the leveraged cost.
Treating a normal approximation percentile as the 99.5% capital figure for a heavy-tailed, low-frequency portfolio.
Fitting severity to raw historical amounts without trending them for inflation, or without correcting the likelihood for censoring at policy limits and truncation at deductibles.
Mixing units: λ per year with severity per claim, or mean in ₹ lakh with E[X²] in ₹², giving answers out by factors of 10⁵.