Reserve models and ruin probabilities

Reducing to the essentials, a collective model of an insurance company’s reserve should include the following components:

• initial reserve r₀, which is the start-up capital of the insurance company.

• income from premium collection by a rate c_t>0 at timet.

• the expenses of claims modelled by a random sum, PNt

i=1X_i at time t, where (N_t)t≥0 is a counting process representing the number of claims and(X_i)i∈N are the non-negative claims sizes commonly assumed to be independent of (N_t)t≥0. Administrative and general costs, such as wages, rent, ect. are usually neglected. The reserve at time t is then

R_t =r₀+ Z t

0

c_tdt−

Nt

X

i=1

X_i. (2.4)

The associated claim surplus process is S_t =

Nt

X

i=1

X_i− Z t

0

c_tdt.

The Cramér-Lundberg model is the standard collective model. It assumes that the premium rate is constant, i.e. c = c_t, that (N_t)t≥0 is a Poisson process as in Definition 1.2 with intensity λ, and that the claim sizes (X_i)_i∈N are i.i.d. and independent of (N_t)t≥0, where we denote the common distribution by F with i’th moment x_i. Further, in accordance with previous notation, let(T_i)i∈N0 be the claim arrival times, and(W_i)_i∈Nthe inter-arrival times which, due to the Poisson assumption, are i.i.d. with an exponential distribution. This is also the model mentioned in Paper B.

A common concern in risk theory is the ruin probability, ψ(r₀) = P(τ_r0 <∞)

Chapter 2. Optimal premium selection as function of the deductible

where τ_r0 = inf{t≥0 :R_t <0|R₀ =r₀} is the time of ruin, i.e. the first time that the reserve becomes negative. The probability of survival is then φ(r₀) = 1−ψ(r₀).

Some is also concerned with the finite time ruin probability ψ(r₀, T) =P(τ_r0 < T), see, e.g., De Vylder and Goovaerts (1988) and Lefévre and Loisel (2008). However, we exclusively consider the infinite-time ruin here. Further, we assume that the net profit condition, c > x₁λ, is satisfied. Otherwise, ruin happens with probability one. Intuitively, the net profit condition says that on average, we should expect the ingoing premium to at least cover the outgoing claims. This is equivalent with having a positive safety loading, i.e. θ =c−λ x₁ > 0. Otherwise, the reserve would drift negatively, eventually causing ruin with certainty.

The Pollaczek-Khinchin formula represents the ruin probability as the tail of a compound geometric distribution,

ψ(r₀) =

1− x₁λ c

X^∞

n=0

x₁λ c

n

F_X,I^∗(n)(r₀), (2.5) where F_X,I is the integrated tail distribution

F_X,I(r₀) = 1 x1

Z r0

0

F(u)du.

From this formula, it follows that the ruin probability is available explicitly in the case of exponentially distributed claim sizes, i.e. F(x) = 1−exp(−bx)if x >0with b > 0. The ruin probability then simplifies to

ψ(r₀) = 1

1 +θexp

− bθ 1 +θr₀

.

Another case where it is possible to get an exact formula is when r₀ = 0, then ψ(0) = (1 +θ)⁻¹.

Implicitly stated above, the cases where the ruin probability can be found explicitly are few. Therefore the focus in the literature has instead been redirected at creating upper bounds for the ruin probability and examining the asymptotic behaviour. We next outline some of the standard results within this regard of ruin theory. For further interest, we refer to Asmussen and Albrecher (2010). We distinguish according to the heavy-tailedness of the claim size distribution in the sense of Section 1.1.6.

Recall from Section 1.1.6 that claims are said to be light-tailed distributed if there exists a ν > 0 such that m_X(ν) < ∞. In this case, the moment generating function of the claim surplus process,

m_St(x) =E[exp(xS_t)] = exp t(λ(m_X(x)−1)−cx) ,

is also finite on(0, ν). Applying the logarithm yields the cumulant generating function κ_St(x) = log(m_St(x)) =t λ(m_X(x)−1)−cx

. (2.6)

Note that κ_St(0) = 0 and has negative slope in this point as κ⁰_S

t(0) =t(λ x₁−c)<0 for t≥0

due to the net profit condition. κ_St(x)is furthermore a convex function since κ⁰⁰_St(x) = tλE[X²exp(xX)]>0 for t≥0.

From the established properties it follows that if there exists a γ > 0 such that κ_St(γ) = 0, or equivalently, such that

mX(γ) = cγ

λ + 1, (2.7)

then it is unique. This γ is known as the adjustment coefficient (or the Lundberg exponent). It features in two essential results in ruin theory.

The first is Lundberg’s inequality from Lundberg (1909) providing an upper bound on the ruin probability.

Theorem 2.1 (Lundberg’s inequality). Assuming that the adjustment coefficient γ exists, then

ψ(r0)≤exp(−γr0).

The second gives the asymptotics of the ruin probability and is due to Cramér (1930).

Theorem 2.2 (Cramer’s ruin bound). Assume that the F has a density and that the adjustment coefficient γ exists, then there is a constant C such that

ψ(r₀)∼Cexp(−γr₀) as r₀ → ∞.

In the proof ofCramer’s ruin bound, it is useful to write the ruin probability as a defective renewal equation,

ψ(r₀) = qF_X,I(r₀) + Z r0

0

ψ(r₀−x)d(qF_X,I(x))

where q = λ x₁/c < 1. We have previously seen this type of integral equation in Section 1.1.3. Expressing the ruin probability as a renewal equation makes it possibly to use renewal theory techniques to find the asymptotic properties.

For heavy-tailed claims,m_X(ν) =∞ for all ν > 0. An adjustment coefficient can therefore not exist, and the analysis above becomes inapplicable. Instead, considering the subexponential subclass of heavy-tailed distributions, Embrechts and Veraverbeke (1982) show that the ruin probability asymptotically behaves as follows.

Theorem 2.3. Assume that x₁ <∞, that the claim size distribution has a density, and that the integrated claim size distribution F_X,I is subexponential. Then the ruin probability satisfies

ψ(r₀)∼ λ x₁

c−λ x₁F_X,I(r₀) as r₀ → ∞.

Chapter 2. Optimal premium selection as function of the deductible

It now becomes apparent how heavy-tailed claims endangers the solvency of the insurance company. Theorem 2.2 states that for light-tailed claims the ruin probability decays exponentially asymptotically, whereas in Theorem 2.3 we see that for heavy-tailed claims, the ruin probability is of the same order as the integrated tail of the claims. Hence, in the heavy-tailed case ruin can be spontaneously caused by one large claim, corresponding to the principle of a single big jump discussed in Section 1.1.6

The fact that first passage times are more easily calculated for diffusion processes has lead to the idea of approximating the reserve by a diffusion. Iglehart (1969) shows how a sequence of risk reserve processes converges weakly to the Brownian motion with drift,

dR^(d)_t =µdt+σdW_t, R^(d)₀ =r₀ (2.8) where µ=c−λ x₁ and σ² =λ x₂. The argument is based on compressing the time scale, such that claims become many and small, meanwhile maintaining the first two moment of the original reserve process. Iglehart (1969) further shows that for a positive drift (corresponding to the net profit condition being satisfied), the ruin probability of the sequence of risk reserves then converges to the ruin probability of the diffusion approximation given by

P(τ_r^(d)

0 <∞) = exp

−2r₀µ σ²

.

where τr^(d)0 = inf{t ≥ 0 : R^(d)_t < 0 | R^(d)₀ = r0} is the time of ruin for the diffusion approximation.

We use this diffusion approximation to model the insurance company’s reserve in Paper B, where the simple nature of the associated ruin probability is convenient in the sense that a closed-form solution is obtainable for the optimal single-customer premium.