Loss distributions, with and without risk sharing: formula sheet

Key formulas

Exponential(λ)

f(x) = λe^(−λx); S(x) = e^(−λx); E[X] = 1/λ; Var(X) = 1/λ²

Constant hazard rate λ. MGF = λ ÷ (λ − t) for t < λ.

Gamma(α, λ)

f(x) = λ^α x^(α−1) e^(−λx) ÷ Γ(α); E[X] = α/λ; Var(X) = α/λ²

MGF = (λ ÷ (λ − t))^α for t < λ. Light-tailed. α = 1 gives the exponential.

Lognormal(μ, σ²)

E[X^k] = exp(kμ + k²σ²/2); E[X] = exp(μ + σ²/2); Var(X) = exp(2μ + σ²) × (exp(σ²) − 1)

μ and σ² are the mean and variance of ln X, not of X. The median is e^μ. The MGF does not exist for t > 0.

Pareto(α, λ)

S(x) = (λ ÷ (λ + x))^α; f(x) = αλ^α ÷ (λ + x)^(α+1); E[X] = λ ÷ (α − 1) for α > 1; Var(X) = αλ² ÷ ((α − 1)²(α − 2)) for α > 2

E[X^k] = k! λ^k ÷ ((α−1)(α−2)...(α−k)) only when α > k. This is the two-parameter form on the support x > 0.

Weibull(c, γ)

S(x) = exp(−c x^γ); f(x) = cγ x^(γ−1) exp(−c x^γ); E[X^k] = Γ(1 + k/γ) ÷ c^(k/γ)

Hazard rate is cγ x^(γ−1). It increases for γ > 1, is constant for γ = 1 and decreases for γ < 1.

Burr(α, λ, γ)

S(x) = (λ ÷ (λ + x^γ))^α; f(x) = αγ λ^α x^(γ−1) ÷ (λ + x^γ)^(α+1); E[X^k] = λ^(k/γ) Γ(k/γ + 1) Γ(α − k/γ) ÷ Γ(α)

The k-th moment exists only for −γ < k < αγ. With γ = 1 it reduces to the Pareto.

Light vs heavy tail test

Light-tailed: M_X(t) exists for some t > 0. Heavy-tailed: it does not.

Exponential, gamma and Weibull with γ ≥ 1 are light-tailed. Lognormal, Pareto and Burr are heavy-tailed. Weibull with γ < 1 is also usually treated as heavy-tailed.

Method of moments

E[X^k; θ] = (1/n) Σ xᵢ^k, for k = 1, ..., number of parameters

Solve the equations for the parameters. For a two-parameter family use k = 1 and 2.

Percentile matching

F(πₚ; θ) = p, where πₚ is the sample 100p-th percentile

Use as many percentiles as parameters. State how you computed the sample percentile.

Log-likelihood, complete data

l(θ) = Σ ln f(xᵢ; θ)

Solve dl/dθ = 0 and check it is a maximum.

Right-censored observation at u

contribution to L = S(u) = 1 − F(u)

Applies when you only know the claim exceeds u, such as a policy limit.

Left-truncated observation at d

contribution to L = f(x) ÷ S(d), for x > d

Applies when claims below the deductible are not recorded.

Exponential MLE

λ̂ = n ÷ Σ xᵢ = 1 ÷ x̄ (rate λ)

With censored data: λ̂ = (number of uncensored claims) ÷ (sum of all amounts, including censored).

Pareto moments (α, λ), f(x) = αλ^α ÷ (λ + x)^(α+1)

E[X] = λ ÷ (α − 1) for α > 1; E[X²] = 2λ² ÷ ((α − 1)(α − 2)) for α > 2

Moments exist only above these values of α.

Gamma (α, λ) moments

E[X] = α ÷ λ; Var(X) = α ÷ λ²

MoM: λ̂ = x̄ ÷ s², α̂ = x̄² ÷ s².

Ordinary deductible, per loss

Y = (X − d)+ = max(X − d, 0)

Payment is zero when X ≤ d. Its distribution has a mass at 0 of F(d).

Limited loss

min(X, u) = X if X ≤ u; u if X > u

Use for a policy limit with no deductible.

Payment with deductible d and loss cover up to u

Y = min(X, u) − min(X, d), for d < u

Maximum payment is u − d.

Expected payment per loss (deductible)

E[(X − d)+] = ∫ from d to ∞ of S(x) dx

S(x) = 1 − F(x) is the survival function. Needs the mean of X to be finite.

Expected limited loss

E[min(X, u)] = ∫ from 0 to u of S(x) dx

Also E[X] = E[min(X, d)] + E[(X − d)+].

Expected payment with deductible and limit

E[min(X, u) − min(X, d)] = ∫ from d to u of S(x) dx

Area under the survival curve between d and u.

Second moment of limited loss

E[min(X, u)²] = ∫ from 0 to u of 2x S(x) dx

Use with the mean to get variance: Var = E[Y²] − (E[Y])².

Per payment versus per loss

E[Y per payment] = E[(X − d)+] ÷ S(d), where S(d) = P(X > d)

Same relation holds for E[Y²]. Mean excess loss e(d) = E[X − d | X > d].

Exponential with rate λ

E[(X − d)+] = e^(−λd) ÷ λ; E[min(X, u)] = (1 − e^(−λu)) ÷ λ

By the lack of memory, X − d given X > d is again exponential with rate λ.

Pareto with parameters α, λ: S(x) = (λ ÷ (λ + x))^α

X − d given X > d is Pareto(α, λ + d); mean per payment = (λ + d) ÷ (α − 1)

Needs α > 1 for the mean and α > 2 for the variance. Variance of Pareto(α, λ) = αλ² ÷ ((α − 1)²(α − 2)).

Franchise deductible

Y = 0 if X ≤ d; Y = X if X > d. E[Y] = E[(X − d)+] + d × S(d)

Pays the full loss once the loss exceeds d.

Claim split

X = Y + Z

Y is the insurer's payment, Z is the reinsurer's payment, for every claim.

Quota share payments

Y = αX, Z = (1 − α)X, with 0 < α < 1

α is the proportion retained by the insurer.

Quota share mean and variance

E(Y) = αE(X); Var(Y) = α²Var(X); E(Z) = (1 − α)E(X); Var(Z) = (1 − α)²Var(X)

Variance scales by the square of the proportion. The coefficient of variation is unchanged.

Excess of loss payments

Y = min(X, M); Z = max(X − M, 0)

M is the retention. Z equals 0 when X ≤ M.

Expected reinsurer payment (all claims)

E(Z) = ∫ from M to ∞ of (x − M) f(x) dx = ∫ from M to ∞ of [1 − F(x)] dx

This averages over all claims, including those where Z = 0.

Insurer's expected payment

E(Y) = E(X) − E(Z) = ∫ from 0 to M of [1 − F(x)] dx

Useful when E(X) is known.

Excess over retention, given a payment

P(Z > z | X > M) = [1 − F(M + z)] ÷ [1 − F(M)]

This gives the distribution of the payment on claims that reach the reinsurer.

Pareto with excess of loss

If X ~ Pareto(a, λ), then X − M | X > M ~ Pareto(a, λ + M)

Same shape a. Scale increases by M. Valid for M ≥ 0.

Exponential with excess of loss

If X ~ Exp(λ), then X − M | X > M ~ Exp(λ)

Memoryless property. Also P(X > M) = e^(−λM).

Pareto mean and moment

E(X) = λ ÷ (a − 1) for a > 1; E(X²) = 2λ² ÷ [(a − 1)(a − 2)] for a > 2

Using the IAI parametrisation with density aλ^a ÷ (λ + x)^(a+1).

Reinsurer's mean on all claims (Pareto)

E(Z) = P(X > M) × E(X − M | X > M) = [λ ÷ (λ + M)]^a × (λ + M) ÷ (a − 1)

Needs a > 1.

Inflated claim

Y = (1 + r)X

r is total inflation over the period. For n years at annual rate i, use (1 + i)ⁿ.

Probability after inflation

P(Y > d) = P(X > d ÷ (1 + r))

Equivalent to using the new distribution of Y with the same d.

Pareto

X ~ Pareto(α, λ) ⇒ Y ~ Pareto(α, (1 + r)λ)

Shape α unchanged. Mean is λ ÷ (α − 1) for α > 1.

Exponential and gamma

X ~ Exp(λ) ⇒ Y ~ Exp(λ ÷ (1 + r)); X ~ Gamma(α, λ) ⇒ Y ~ Gamma(α, λ ÷ (1 + r))

Rate parameter is divided. Scale (1 ÷ λ) is multiplied.

Lognormal

X ~ LN(μ, σ²) ⇒ Y ~ LN(μ + ln(1 + r), σ²)

σ is unchanged.

Mean and variance

E[Y] = (1 + r)E[X]; Var(Y) = (1 + r)² Var(X)

Standard deviation scales by (1 + r). The coefficient of variation is unchanged.

Expected payment above a fixed deductible

E[(Y − d)+] = (1 + r) × E[(X − d ÷ (1 + r))+]

Use this to price excess of loss cover after inflation with a fixed d.

Quick revision

• Deductible d: the insurer pays Y = max(X − d, 0). Per loss, E[Y] = ∫ from d to ∞ of S(x) dx.
• Policy limit M: the insurer pays min(X, M). E[min(X, M)] = ∫ from 0 to M of S(x) dx.
• Excess of loss with retention M: the insurer pays min(X, M) and the reinsurer pays max(X − M, 0). The two add to X.
• Proportional reinsurance with retention α: the insurer pays αX and the reinsurer pays (1 − α)X.
• Proportional reinsurance scales the mean by α and the variance by α². It does not change the shape of the distribution.
• Exponential: mean 1/λ, and the memoryless property means the excess over d is again exponential with the same λ.
• Pareto and lognormal are heavy tailed. The exponential has a light tail compared with them.
• Method of moments: equate sample moments to the model moments and solve for the parameters.
• MLE: write the log-likelihood, set the derivative to zero, solve, then check the second derivative.
• Inflation by a factor (1 + i) multiplies X by that factor. For a scale family, the scale parameter is multiplied and the shape parameter is unchanged.
• With a fixed deductible, inflation raises the insurer's expected payment by more than the inflation rate on the ground-up claim.
• Always state whether your answer is per loss or per payment.

Common mistakes

• Using the wrong parameterisation, for example treating the Pareto as having support x > λ. Fix: Use the form in the IAI Tables. Check the survival function in the Tables before you start. State the form you are using.
• Treating lognormal μ and σ² as the mean and variance of X. Fix: Remember that μ and σ² belong to ln X. Convert with E[X] = exp(μ + σ²/2) and the variance formula.
• Using f(x) for a censored claim instead of S(u). Fix: Ask for each data point: do I know the exact value, or only that it is above a limit? Use f(x) for exact and S(u) for censored.
• Forgetting to divide by S(d) for claims truncated at a deductible. Fix: If small claims are not observed, every observed claim is conditional on X > d. Use f(x) ÷ S(d).
• Using E[X] − d as the expected payment. Fix: Use E[(X − d)+] = ∫ from d to ∞ of S(x) dx. It is larger than E[X] − d whenever P(X < d) > 0.
• Mixing up per-loss and per-payment means. Fix: Read the wording. Per loss includes zero payments. Per payment divides by S(d). Write the label next to each answer.
• Using α instead of α² for the variance under quota share. Fix: Variance is in squared units. Var(αX) = α²Var(X). The standard deviation scales by α.
• Writing the reinsurer's payment as X − M for every claim. Fix: Write Z = max(X − M, 0). Say whether you are conditioning on X > M.
• Inflating the deductible or retention along with the claims Fix: Read the wording. A fixed rupee deductible stays fixed. Only change it if the question says it is indexed.
• Changing the shape parameter of the Pareto or gamma Fix: The shape (α) never changes. Multiply the scale λ for the Pareto. Divide the rate λ for the gamma and exponential.

Exam tips

• Write the survival function first. Most tail and exceedance questions are one line once S(x) is written.
• Always state the condition for a moment to exist when the question uses a Pareto or Burr. Examiners give marks for it.
• In discussion questions, link each choice of distribution to a data feature: skewness, extreme outliers, hazard behaviour. Do not just list the distributions.
• In MCQs, check whether the parameterisation in the question matches the Tables before you calculate.
• For computer-based work, show the formula and the parameter values you used before quoting any numerical result.
• Write the likelihood contribution for each data type before doing any algebra. Marks are often given for the correct contributions even if the algebra slips.
• Always state the parameterisation you use for Pareto, gamma and exponential, and keep it consistent. The Formulae and Tables book may use a different form from your notes.
• In Paper B (R), know how to write the negative log-likelihood as a function and minimise it with optim or nlm. Give starting values and check convergence.