Surplus Process and Ruin Basics: Ruin Probability Explained

Understand Surplus Process and Ruin Basics

Key rules to remember

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.

How to solve Surplus Process and Ruin Basics questions

1. 1Write down the model: u, c, claim arrival process (rate λ) and claim size distribution, with E[X] and, if needed, E[X²].
2. 2Check the premium: compute λE[X] and compare with c. Find θ from c = (1 + θ)λE[X] and confirm θ > 0.
3. 3Write the surplus U(t) = u + ct − S(t), or U_n in discrete time, using the stated model.
4. 4Define ruin exactly as the question asks: continuous or discrete, finite horizon t or infinite time, and whether ruin means U < 0.
5. 5Express the required probability in terms of T, such as P(T ≤ t) or P(T < ∞), or in terms of U_n for discrete checks.
6. 6Compute what is asked: moments of S(t), expected surplus, loading, or a probability for simple cases such as ruin at the first claim.
7. 7State your assumptions and conclude in words, including the direction of any effect on ruin probability.

Quickest way: Loading check and first-claim shortcut

Common mistakes in Surplus Process and Ruin Basics

• Using Var[X] instead of E[X²] in Var[S(t)].

  Students recall the general compound formula and forget that the Poisson case simplifies.

  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.

  The word loading is also used in pricing premiums generally.

  Fix: Here c = (1 + θ)λE[X]. θ = c ÷ (λE[X]) − 1, not (c − λE[X]) ÷ c.

• Saying ruin is impossible when θ > 0.

  Students confuse positive expected drift with a guarantee.

  Fix: Positive loading only means ruin is not certain. ψ(u) is still positive because a run of large claims can occur.

• Mixing up finite-time and infinite-time ruin probabilities.

  Both use the symbol ψ and the same surplus process.

  Fix: Write ψ(u, t) = P(T ≤ t) and ψ(u) = P(T < ∞). Remember the finite-time value is never larger.

• Assuming discrete-time ruin is the same as continuous-time ruin.

  Both definitions look similar.

  Fix: Discrete checks can miss dips below zero between check dates. So discrete-time ruin probability ≤ continuous-time ruin probability.

• Letting ruin occur between claims in the continuous model.

  Students think the surplus can fall under zero while only premiums are arriving.

  Fix: Between claims the surplus rises at rate c. Ruin can happen only at a claim instant.

Worked examples