Ruin theory: formula sheet

Key formulas

Surplus process

U(t) = u + ct − S(t)

u is initial surplus, c is premium income rate, S(t) is aggregate claims up to time t.

Aggregate claims

S(t) = X₁ + X₂ + ... + X_N(t), with S(t) = 0 if N(t) = 0

N(t) is Poisson with mean λt. The Xᵢ are i.i.d. and independent of N(t).

Mean and variance of S(t)

E[S(t)] = λt·E[X]; Var[S(t)] = λt·E[X²]

Note the variance uses E[X²], not Var[X]. This holds for compound Poisson.

Premium loading

c = (1 + θ)λE[X]

θ is the relative security loading. Net profit condition: θ > 0, that is c > λE[X].

Expected surplus

E[U(t)] = u + (c − λE[X])t = u + θλE[X]t

Grows linearly when θ > 0.

Time of ruin

T = inf{t > 0 : U(t) < 0}

T = ∞ if the surplus never goes below zero.

Infinite-time ruin probability

ψ(u) = P(T < ∞)

Decreases as u increases and as θ increases.

Finite-time ruin probability

ψ(u, t) = P(T ≤ t)

Non-decreasing in t, and ψ(u, t) → ψ(u) as t → ∞.

Discrete-time surplus and ruin

U_n = u + nc − S_n; ruin if U_n < 0 for some n

Checks the surplus only at times 1, 2, 3, ... so ruin probability is no larger than in continuous time.

Inter-arrival times

Time between claims ~ Exponential(λ), mean 1/λ

Follows from Poisson claim arrivals.

Surplus process

U(t) = u + ct − S(t)

S(t) is the compound Poisson total claims with Poisson rate λ.

Premium with loading

c = (1 + θ) λ E[X]

θ is the relative security loading. Need θ > 0 for R to exist.

Lundberg equation

λ M_X(r) = λ + cr, equivalently λ(M_X(r) − 1) = cr

R is the unique positive root. r = 0 is always a trivial root, so ignore it.

Lundberg's inequality

ψ(u) ≤ e^(−Ru)

Valid for u ≥ 0 in the classical compound Poisson model.

Exponential claims, mean 1/β

R = β − λ/c = βθ/(1 + θ)

Here the claim density is βe^(−βx) and c = (1 + θ)λ/β.

Exact ruin for exponential claims

ψ(u) = (1/(1 + θ)) e^(−Ru)

The bound e^(−Ru) is correct in the exponent but overstates the constant.

Exponential MGF

M_X(r) = β/(β − r) for r < β

Root must satisfy r < β.

Premium with loading

c = (1+θ)λμ

θ is the relative security loading. Net claims rate is λμ. You need θ > 0 for ruin probability to be below 1.

Ruin probability at zero surplus

ψ(0) = 1 ÷ (1+θ)

Holds for the compound Poisson model with any claim size distribution that has finite mean, in infinite time.

Exponential claims, exact ruin probability

ψ(u) = (1 ÷ (1+θ)) × exp(−θu ÷ ((1+θ)μ))

Claims exponential with mean μ. Equivalent form: ψ(u) = (λμ ÷ c) × e^(−Ru).

Adjustment coefficient, exponential claims

R = θ ÷ ((1+θ)μ)

With rate parameter β = 1/μ, this is R = β − λ/c.

General adjustment coefficient equation

λ + cR = λ M_X(R), R > 0

M_X(R) = E[e^(RX)] is the moment generating function of a claim. Take the positive root.

Lundberg's inequality

ψ(u) ≤ e^(−Ru)

A bound, valid for u ≥ 0 when R exists.

Cramér-Lundberg approximation

ψ(u) ≈ C e^(−Ru)

Asymptotic for large u. For exponential claims C = 1 ÷ (1+θ) and it is exact for all u ≥ 0.

Discrete-time surplus

U(t) = u + c·t − S(t), where S(t) = X₁ + X₂ + … + X_t

Checked at t = 1, 2, 3, … only. Claims X_i are independent and identically distributed in the basic model.

Finite-time ruin probability

ψ(u, n) = P(U(t) < 0 for some t in {1, 2, …, n}) = 1 − φ(u, n)

Ruin is strictly below zero. State this convention in your answer.

Boundary conditions

φ(u, 0) = 1 for u ≥ 0, and φ(u, n) = 0 for u < 0

Use the second one whenever the surplus after a period is negative.

Survival recursion

φ(u, n) = Σ over x of P(X = x) · φ(u + c − x, n − 1)

For discrete claims. For continuous claims the sum becomes an integral. Terms with u + c − x < 0 are zero.

Simulated estimate of ruin probability

ψ̂ = (number of ruined paths) ÷ N

N is the number of simulated paths.

Standard error and approximate 95% confidence interval

SE = √(ψ̂(1 − ψ̂) ÷ N); interval: ψ̂ ± 1.96 × SE

Uses the normal approximation, which is fine when N is large and ψ̂ is not very close to 0 or 1.

Simulations needed for a target standard error

N = p(1 − p) ÷ SE²

Use a rough guess of p, or p = 0.5 for the worst case.

Inverse transform for a discrete claim

Generate V from U(0,1). Pick the claim whose cumulative probability range contains V.

Fix the rule first, for example V < 0.6 gives claim 1, otherwise claim 4.

Original adjustment coefficient equation

λ + cR = λ M_X(R), with c = (1 + θ)λE[X]

R is the smallest positive root. It is the equation with no reinsurance.

Lundberg's inequality

ψ(u) ≤ e^(−Ru)

Valid when a positive R exists. It is an upper bound, not the exact ruin probability.

Proportional reinsurance: net premium rate

c_α = c − (1 + ξ)(1 − α)λE[X]

α is the proportion retained by the insurer, and ξ is the reinsurer's loading.

Proportional reinsurance: equation for R

λ + c_α R = λ M_X(αR)

The retained claim αX has mgf M_X(αR). Solve for the positive root.

Proportional reinsurance: condition for a positive R

c_α > αλE[X], that is θ > ξ(1 − α)

If ξ > θ, this requires α > 1 − θ/ξ. If ξ ≤ θ, it holds for every α > 0.

Excess of loss: net premium rate

c_M = c − (1 + ξ)λE[(X − M)+]

M is the retention. For exponential claims with mean m, E[(X − M)+] = m e^(−M/m).

Excess of loss: equation for R

λ + c_M R = λ E[e^(R·min(X, M))]

The retained claim is min(X, M). It is bounded above by M.

Exponential claims, mean m, net premium c' and retained mean m'

R = (c' − λm')/(c'·m'), when λ = 1 per unit time

This is a shortcut for exponential claims, valid for claim rate λ = 1 and exponential retained claims. Here c' and m' are the retained values.

Quick revision

• Surplus: U(t) = u + ct − S(t), where S(t) is aggregate claims up to time t.
• Premium rate: c = (1 + θ)λE[X], with θ > 0 the relative security loading.
• Ruin probability ψ(u) is the probability that surplus ever falls below zero, starting from surplus u.
• Adjustment coefficient R is the positive solution of λ + cR = λM_X(R).
• Lundberg's inequality: ψ(u) ≤ e^(−Ru) for u ≥ 0.
• Larger R means a smaller bound on ruin, so lower risk.
• In the compound Poisson model, ψ(0) = 1/(1 + θ), whatever the claim size distribution.
• Exponential claims with mean μ: R = θ / ((1 + θ)μ) and ψ(u) = (1/(1 + θ)) e^(−Ru).
• Increasing θ or starting surplus u lowers ψ(u).
• Simulation estimates ψ(u) as the proportion of simulated paths that reach ruin; more paths give a smaller standard error.
• After reinsurance, use retained claims and net premium (premium less reinsurance cost), then re-solve for R.
• State your assumptions: independent claims, Poisson arrivals, constant premium rate, and the time horizon used.

Common mistakes

• Using Var[X] instead of E[X²] in Var[S(t)]. Fix: For compound Poisson, Var[S(t)] = λt·E[X²]. Compute E[X²] = Var[X] + (E[X])² first.
• Treating loading as a percentage of premium rather than of expected claims. Fix: Here c = (1 + θ)λE[X]. θ = c ÷ (λE[X]) − 1, not (c − λE[X]) ÷ c.
• Accepting r = 0 as the answer. Fix: Divide by r or state clearly that you need the positive root.
• Using the wrong premium rate c. Fix: Compute c = (1 + θ) λ E[X] and check the units against λ.
• Using c as the loading, for example writing ψ(0) = 1 ÷ c. Fix: Always compute θ = c ÷ (λμ) − 1 first. ψ(0) = 1 ÷ (1+θ) = λμ ÷ c.
• Writing R = θ ÷ μ for exponential claims. Fix: Remember R = θ ÷ ((1+θ)μ). Check by R = 1/μ − λ/c.
• Treating surplus of exactly zero as ruin. Fix: Use ruin as surplus < 0 unless the question says otherwise. Say so at the start of your answer.
• Forgetting to add the premium before subtracting the claim in each period. Fix: Always write the next surplus as u + c − x and check it against zero.
• Using the original premium c in the equation after reinsurance. Fix: Always compute the net premium c_α or c_M first. Write it down before forming the equation.
• Using M_X(R) instead of M_X(αR) for proportional reinsurance. Fix: The retained claim is αX, so its mgf at R is M_X(αR). Check that the left side uses c_α.

Exam tips

• Write the definitions with exact symbols: T = inf{t > 0 : U(t) < 0}, ψ(u) = P(T < ∞) and ψ(u, t) = P(T ≤ t). Marks are often for the definition alone.
• Compute λE[X] first in any numerical question. It gives θ, expected claims and expected surplus quickly.
• In multiple-choice questions, watch for the variance trap: Var[S(t)] = λt·E[X²], not λt·Var[X].
• State your assumptions: claims i.i.d., independent of arrivals, Poisson arrivals, premiums continuous and constant, and θ > 0.
• In written answers, finish with a one-line interpretation, such as how ruin probability changes as u or θ rises. Ruin probability falls as either rises.
• Always state the model assumptions: compound Poisson claims, constant premium rate, independent claims and u ≥ 0.
• Show the substitution into the Lundberg equation before you simplify. Method marks sit there.
• Check that the loading is positive and that your root lies inside the MGF range.