Credibility theory: Bayesian and empirical Bayes credibility premiums: formula sheet

Key formulas

Credibility premium

P = Z × X̄ + (1 − Z) × M

X̄ is the group's own mean experience, M is the manual or prior estimate, 0 ≤ Z ≤ 1.

Full credibility condition (mean of n i.i.d. observations)

P(|X̄ − μ| ≤ k·μ) ≥ p

Using the normal approximation, this holds when n ≥ (y ÷ k)² × (σ² ÷ μ²), where y satisfies Φ(y) = (1 + p) ÷ 2.

Full credibility standard for claim numbers (Poisson)

n_F = (y ÷ k)² (expected number of claims)

Applies when the number of claims is Poisson with mean λ, so λ_F = (y ÷ k)². Here the standard is in expected claims.

Full credibility standard for aggregate claims (compound Poisson)

λ_F = (y ÷ k)² × (1 + CV²), where CV = σ_Y ÷ μ_Y of the claim size

Equivalently λ_F = (y ÷ k)² × E[Y²] ÷ (E[Y])². The standard is in expected number of claims.

Partial credibility (square root rule)

Z = √(n ÷ n_F), capped at 1

n and n_F must be in the same units, such as number of claims or exposure years.

Common normal values for y

p = 90%: y = 1.645; p = 95%: y = 1.960; p = 99%: y = 2.576

y is the standard normal point with Φ(y) = (1 + p) ÷ 2, a two-sided probability.

Bayes' theorem for parameters

π(θ | x) = f(x | θ) π(θ) ÷ ∫ f(x | θ) π(θ) dθ

For a discrete θ, replace the integral with a sum. The denominator does not depend on θ.

Proportionality form

π(θ | x) ∝ f(x | θ) × π(θ)

Drop any factor without θ, then identify the distribution.

Likelihood for a random sample

f(x | θ) = Π f(xᵢ | θ), i = 1 to n

Assumes the observations are independent given θ.

Squared error loss

L(θ̂, θ) = (θ̂ − θ)² → θ̂ = E[θ | x]

The Bayes estimate is the posterior mean.

Absolute error loss

L(θ̂, θ) = |θ̂ − θ| → θ̂ = median of π(θ | x)

Solve F(θ̂ | x) = 0.5 using the posterior distribution function.

All-or-nothing loss

θ̂ = mode of π(θ | x)

For a 0-1 loss on a continuous θ, this is the limiting case; the mode is the Bayes estimate.

Poisson data, Gamma prior

Xᵢ | λ ~ Poisson(λ), λ ~ Gamma(α, β) → λ | x ~ Gamma(α + Σxᵢ, β + n)

Here β is the rate, so the prior density is ∝ λ^(α−1) e^(−βλ) and the mean is α ÷ β.

Binomial data, Beta prior

Xᵢ | θ ~ Bin(m, θ), θ ~ Beta(a, b) → θ | x ~ Beta(a + Σxᵢ, b + nm − Σxᵢ)

Beta mean is a ÷ (a + b). Beta mode is (a − 1) ÷ (a + b − 2) when a, b > 1.

Exponential data, Gamma prior

Xᵢ | λ ~ Exp(λ), λ ~ Gamma(α, β) → λ | x ~ Gamma(α + n, β + Σxᵢ)

Here λ is the rate of the exponential.

Normal mean, known variance

Xᵢ | μ ~ N(μ, σ²), μ ~ N(μ₀, σ₀²) → 1 ÷ σ₁² = 1 ÷ σ₀² + n ÷ σ², μ₁ = σ₁² (μ₀ ÷ σ₀² + n x̄ ÷ σ²)

Posterior is normal. Mean and median and mode all equal μ₁.

Credibility form of the posterior mean (Poisson-Gamma)

E[λ | x] = Z x̄ + (1 − Z) (α ÷ β), Z = n ÷ (n + β)

Shows the estimate as a weighted average of data mean and prior mean.

Bayesian premium (predictive mean)

E[X_{n+1} | x] = ∫ μ(θ) f(θ | x) dθ

Use a sum for a discrete θ. μ(θ) = E[X | θ]. X_{n+1} is independent of the past given θ.

Posterior density

f(θ | x) ∝ f(θ) × Π f(xᵢ | θ)

Drop every factor not involving θ, then recognise the distribution from its kernel.

Credibility form

P = Z × x̄ + (1 − Z) × E[μ(θ)]

Valid exactly only when the posterior mean of μ(θ) is linear in the data, as in conjugate cases above.

Poisson-gamma model

X | λ ~ Poisson(λ), λ ~ Gamma(α, β) (mean α ÷ β). Posterior: Gamma(α + Σxᵢ, β + n)

Here β is the rate parameter. Check which parametrisation the question uses.

Poisson-gamma premium and Z

P = (α + Σxᵢ) ÷ (β + n) = Z x̄ + (1 − Z)(α ÷ β), with Z = n ÷ (n + β)

So k = β. Z is found by matching the coefficient of x̄.

Normal-normal (known variance)

X | θ ~ N(θ, σ²), θ ~ N(μ, s²). Z = n ÷ (n + σ² ÷ s²)

Posterior mean = Z x̄ + (1 − Z) μ.

Binomial-beta (m trials each)

X | q ~ Bin(m, q), q ~ Beta(a, b). Posterior mean of q = (a + Σxᵢ) ÷ (a + b + nm)

Premium per trial is this value, so Z = nm ÷ (a + b + nm).

Hypothetical mean and process variance

m(θ) = E[X | θ]; s²(θ) = Var(X | θ)

Both are functions of θ, so they are random variables before θ is known.

Overall mean

μ = E[m(θ)]

This is the collective premium, the mean claim of the whole portfolio.

EPV

EPV = E[s²(θ)] (often written v)

Average of the process variances over the distribution of θ.

VHM

VHM = Var(m(θ)) (often written a)

Variance of the hypothetical means.

Parameter k

k = EPV ÷ VHM

Larger k means less weight on own experience.

Credibility factor

Z = n/(n + k) = n·VHM / (n·VHM + EPV)

n is the number of observations (years) for the risk.

Credibility premium

P = Z·X̄ + (1 − Z)·μ

X̄ is the risk's sample mean over n years.

Total variance link

Var(X) = EPV + VHM

Follows from the law of total variance for a single observation.

Variance of sample mean

Var(X̄) = VHM + EPV ÷ n

Useful in the least squares derivation.

Model assumptions

E[X_ij | θ_i] = μ(θ_i); Var(X_ij | θ_i) = σ²(θ_i) ÷ P_ij

P_ij is the known exposure. Given θ_i, observations are independent.

Parameters

m = E[μ(θ)]; s² = E[σ²(θ)]; a = Var[μ(θ)]; k = s² ÷ a

k is the exposure at which Z = 50%.

Total exposure and weighted mean for risk i

P_i = Σ_j P_ij; X̄_i = Σ_j P_ij X_ij ÷ P_i

Weight each year by its exposure, not equally.

Credibility factor

Z_i = P_i ÷ (P_i + k)

Increases with total exposure P_i and falls as k rises.

Credibility premium (per unit of exposure)

Z_i X̄_i + (1 − Z_i) m

For total claims next year, multiply by next year's exposure.

Empirical Bayes: overall mean

m̂ = X̄ = Σ_i Σ_j P_ij X_ij ÷ P, where P = Σ_i P_i

Weighted mean over all risks and years.

Empirical Bayes: estimate of s²

ŝ² = Σ_i Σ_j P_ij (X_ij − X̄_i)² ÷ Σ_i (n_i − 1)

n_i is the number of years for risk i. Needs at least one risk with n_i ≥ 2.

Empirical Bayes: estimate of a

â = [Σ_i P_i (X̄_i − X̄)² − (N − 1) ŝ²] ÷ [P − Σ_i P_i² ÷ P]

N is the number of risks. If â is negative, set it to 0, which gives Z = 0.

Model 1: overall mean estimate

Ê[m(θ)] = X̄ = (1 ÷ Nn) Σi Σj Xij

N risks, n years each, equal exposure. X̄i is the mean of risk i.

Model 1: EPV estimate

Ê[s²(θ)] = (1 ÷ N) Σi [ (1 ÷ (n − 1)) Σj (Xij − X̄i)² ]

Average of the sample variances of each risk, each with divisor n − 1.

Model 1: VHM estimate

V̂ar[m(θ)] = (1 ÷ (N − 1)) Σi (X̄i − X̄)² − Ê[s²(θ)] ÷ n

If the result is negative, set it to 0.

Model 1: credibility factor and premium

Z = n ÷ (n + k), k = EPV ÷ VHM; premium = Z X̄i + (1 − Z) X̄

Z = n ÷ (n + EPV ÷ VHM) is the same thing.

Model 2: weighted means

Pi = Σj Pij; P = Σi Pi; X̄iP = Σj Pij Xij ÷ Pi; X̄PP = Σi Pi X̄iP ÷ P

Xij is claims per unit exposure. X̄PP is the estimate of E[m(θ)].

Model 2: EPV estimate

ŝ² = Σi Σj Pij (Xij − X̄iP)² ÷ Σi (ni − 1)

With equal years n, the divisor is N(n − 1).

Model 2: VHM estimate

σ̂² = [ Σi Pi (X̄iP − X̄PP)² − (N − 1) ŝ² ] ÷ [ P − Σi Pi² ÷ P ]

Set to 0 if negative.

Model 2: credibility factor and premium

Zi = Pi ÷ (Pi + ŝ² ÷ σ̂²); premium per unit exposure = Zi X̄iP + (1 − Zi) X̄PP

Multiply by next year's exposure to get the total premium.

Quick revision

• Credibility premium = Z × own experience + (1 − Z) × collective mean, with 0 ≤ Z ≤ 1.
• Z increases as the amount of data increases.
• Bühlmann: Z = n ÷ (n + k), where k = E[s²(θ)] ÷ Var[m(θ)].
• μ = E[m(θ)], where m(θ) = E[X | θ] and s²(θ) = Var[X | θ].
• Total variance of X = E[s²(θ)] + Var[m(θ)].
• Bühlmann-Straub: let P_i be the total exposure for risk i. Then Z_i = P_i ÷ (P_i + k), with k = E[s²(θ)] ÷ Var[m(θ)], where s²(θ) is the process variance per unit of exposure. X̄_i is the exposure-weighted mean claim per unit of exposure for risk i (total claims ÷ P_i).
• Poisson-Gamma: posterior is Gamma(α + Σx, β + n), and Z = n ÷ (n + β) with a rate β.
• The Bayesian premium equals the Bühlmann premium when the posterior mean is linear in the data, as for Poisson-Gamma, Normal-Normal with known variance, and Binomial-Beta. Binomial-Beta needs a fixed number of trials per observation. In general, the Bühlmann premium is the best linear approximation to the Bayesian premium in the least-squares sense.
• Limited fluctuation, for Poisson claim numbers: the full credibility standard is n_F = (y ÷ k)² expected claims, where P(|N(0,1)| ≤ y) = (1 + p) ÷ 2. Partial credibility is Z = min(1, √(n ÷ n_F)), with n in expected claims.
• The letter k means different things: in limited fluctuation it is the tolerance (maximum deviation), and in Bühlmann it is the variance ratio E[s²(θ)] ÷ Var[m(θ)].
• Empirical Bayes: use sample variances with divisor n − 1. Set a negative Var[m(θ)] estimate to zero, which gives Z = 0 and a premium equal to the overall mean X̄ for every risk.
• A large k in Bühlmann means the data is noisy relative to the spread between risks, so Z is small.
• Always state what the premium is for, such as the next year's expected claim per unit of exposure.

Common mistakes

• Using the wrong y, such as 1.645 for 95%. Fix: The condition is two-sided. Use Φ(y) = (1 + p) ÷ 2, so 95% gives 1.960 and 90% gives 1.645.
• Forgetting to square: writing the standard as y ÷ k. Fix: Write n_F = (y ÷ k)² every time. For p = 90% and k = 5%, it is (1.645 ÷ 0.05)² = 1,082.41, not 32.9.
• Mixing up rate and scale in the Gamma distribution. Fix: Write the density ∝ λ^(α−1) e^(−βλ) beside your working. The mean is α ÷ β for a rate β. Check the question's definition before you update.
• Using the wrong number of failures in the Beta update. Fix: The second parameter increases by the number of failures. For n Bernoulli trials with s successes it is b + n − s.
• Using the posterior mean of θ instead of the posterior mean of μ(θ). Fix: Always write μ(θ) = E[X | θ] first. For exponential with mean 1 ÷ θ you need E[1 ÷ θ | x], not E[θ | x].
• Mixing up the gamma rate and scale parameters. Fix: Check the stated mean. If the prior mean is α ÷ β, β is a rate and k = β. If it is αθ, convert.
• Using the variance of all claims instead of EPV Fix: EPV is the average of within-risk variances. Var(X) = EPV + VHM, so subtract VHM if you only have the total.
• Computing VHM as the average of m(θ) squared, or forgetting to subtract μ² Fix: VHM = E[m²] − μ². Check that it is positive.
• Using the simple average of yearly claims instead of the exposure-weighted mean. Fix: Always compute X̄_i = Σ P_ij X_ij ÷ P_i. Equal weights only apply when all P_ij are equal.
• Using Z = n ÷ (n + k) with n as the number of years. Fix: In Buhlmann-Straub, use total exposure: Z = P_i ÷ (P_i + k).

Exam tips

• Write y and the line Φ(y) = (1 + p) ÷ 2 at the start. Marks are often given for the correct y.
• State your units for n_F: claims, policy-years or amount. Examiners look for consistent units.
• Always say what the formula assumes, such as Poisson claim numbers and the normal approximation.
• In written answers, comment on weaknesses of limited fluctuation: arbitrary p and k, and no use of the manual premium's variance. This links to Bühlmann credibility.
• In MCQs, check the answer lies between X̄ and M. This eliminates wrong options quickly.
• Read the loss function first. Many marks are lost by giving the mean when the question wants the median or the mode.
• State the parametrisation of the Gamma you use. Write 'rate β, mean α ÷ β' in your answer so the marker can follow your update.
• In written answers show the proportionality step: prior × likelihood, drop constants, identify the kernel. Marks are given for the method as well as the final distribution.