How much we gain by surplus-dependent premiums -- asymptotic analysis of ruin probability

In this paper, we build on the techniques developed in Albrecher et al. (2013), to generate initial-boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and an exponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations developed in Fedoryuk (1993), we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are {\it linearly} dependent on reserves, representing for instance returns on risk-free investments of the insurance capital, we firstly derive explicit formulas for the ruin probabilities, from which we can easily determine their asymptotics, only to match the ones obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve own by the insurance company.


Introduction
Insurance companies maintain solvency via careful design of premiums rates. The premiums rates are primarily based on the claims history and carefully adjusted to evolving factors such as the number of customers and/or the returns from investments in the financial market. Collective risk models, introduced by Lundberg and Cramér, describe the evolution of the surplus of an insurance business with constant premiums rate, for the simplicity of arguments. This model, a compound Poisson process with drift, is referred to in the actuarial mathematics literature as the Cramér-Lundberg model. However, in practical situations, risk models with surplus-dependent premiums capture better the dynamics of the surplus of an insurance company. Lin and Pavlova (2006) advised for a lower premium for higher surplus level to improve competitiveness, whereas a higher premium is needed for lower surplus level to reduce the probability of ruin.
Among surplus-dependent premiums, risk models with risky investments have been widely analyzed (see e.g. Paulsen (1993); Paulsen and Gjessing (1997); Frolova et al. (2002); Albrecher et al. (2012)). See Paulsen (1998Paulsen ( , 2008 for surveys on the topic. The special case of risk models with linearly dependent premiums can be interpreted as models with riskless investments, since the This work is partially supported by Polish National Science Centre Grant No. 2018/29/B/ST1/00756, 2019-2022. volatility of return on investments or the proportion of the capital invested in the risky asset is zero. Under this scenario, exact expressions of the ruin probability are derived for compound Poisson risk models with interest on surplus and exponential-type upper bounds for renewal risk models with interest (see Dickson (2002, 2003)). Cheung and Landriault (2012) investigate risk models with surplus-dependent premiums with dividend strategies and interest earning as a special case.
Throughout this paper, we build on the method developed in Albrecher et al. (2013) to extend the derivation of ruin probabilities to surplus-dependent premiums risk models with Erlang distributions (claim sizes or interarrival times). Recall from Albrecher et al. (2013), the risk model with surplusdependent premiums is described by where U (t) denotes the surplus at time t, and p(·) is the premium rate at time t, a positive function of the current surplus U (t). When p(.) is constant, this model reduces to the classical collective risk model, see Asmussen and Albrecher (2010). As in classical collective risk theory, ruin defines the first time the surplus becomes negative. For T u , the time of ruin, given by the probability of ruin with initial value u is defined as We focus on calculating ruin probabilities under Erlang claims and arrivals. Previously, Willmot (2007) considered mixed Erlang claim size class when examining various properties associated with renewal risk processes with constant premium rates. Furthermore, Willmot and Woo (2007) applied Erlang mixture to the claim size distribution when discussing the application of ruin-theoretic quantities. Various studies of ruin probabilities focus on risk model with interclaim times being Erlang(n) distributed (see Li and Garrido (2004); Gerber and Shiu (2005); Li and Dickson (2006)) and Erlang(2) distributed (see Dickson and Hipp (2001); Tsai and Sun (2004); Dickson and Li (2010)).
We use an algebraic approach to derive the equations satisfied by the ruin probabilities, similar to the one from Albrecher et al. (2010), and further perform an asymptotic analysis of their solutions. We even solve them explicitely in a few instances. For perspective, Albrecher et al. (2010) introduced an algebraic approach to study the Gerber-Shiu function, and derived a linear ordinary differential equation (ODE) with constant coefficients for claims distribution with rational Laplace transform. Later in 2013, they extended this approach to an ODE with variable coefficients for surplus-dependent premiums risk models. Using method based on boundary value problems and Green's operators, they derived the explicit form of the ruin probability in the classical model with exponential claim sizes. Albrecher et al. (2013) extended the method to surplus-dependent premium models with exponential arrivals, for which they derived exact and asymptotic results for a few premium functions, when the claims were exponentially distributed. Here we extend to renewal models and Erlang claims.
The novelty of the paper consists on the explicit asymptotic analysis performed for reserve dependent premium with Erlang distributed generic claim sizes or Erlang distributed generic interarrival times. We separate the analysis between p(∞) = c and p(∞) = ∞ and use the approximation theory of solutions of linear ordinary differential equations developed in Fedoryuk (1993) to conclude the asymptotics of the ruin probabilities when initial reserves tend to infinity.
Among the premium functions exploding at infinity, i.e. p(∞) = ∞, we consider the linear premium p(u) = c + εu, in which ε can be interpreted as the interest rate on the available surplus. linear premiums can be interpreted as investment of the company in bonds or risk-free assets. When considering premiums that are linearly dependent on reserves, we firstly derive explicit formulas for the ruin probabilities, using confluent geometric functions and their corresponding ODEs. From these exact expressions we can easily determine their asymptotics, only to match the ones obtained for general premiums dependent on reserves.
We show that when the investments are made on risk-free assets only, as bonds or treasury bills, the solvency is improved. We will look at the improvements on solvency when such investments are made, by analyzing the insurance risk models with or without investment returns, for claims and claim arrivals that are exponential or Erlang distributed. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve own by the insurance company.
We consider two cases of premium functions: P1. the premium function behaves like a constant at infinity for c > 0 or P2. the premium function explodes at infinity, p(∞) = ∞ as The first case is satisfied by the rational and exponential premium functions. The second case is satisfied by the linear and quadratic premium functions.
The paper is organized as folows. In Section 2, we introduce the Gerber-Shiu function and present the derivation of the boundary value problem for them in models with premium dependent on reserves and times and claims from distributions with rational Laplace transforms. We recall the results for ruin probabilities, in models with premiums dependent on reserves, general and linear premiums, when both inter-arrivals and claim sizes are exponentially distributed. In Sections 3 and 4, we perform the asymptotic analysis for the ruin probabilities for exponential and Erlang (2) distributed claim sizes and interarrival times, alternatively, for models with premiums dependent on reserves. In each section, for linear premiums, the exact ruin probabilities are derived and the asymptotics confirmed with those obtained for general premiums. Section 5 is dedicated to comparing the asymtotic results, highlighting the gain generated, as in higher solvency, when dynamically adjusting the premium rates to surplus. Conclusions are given in Section 6.

Ruin probabilities -method
Ruin probability is sometimes seen as a particular case of the Gerber-Shiu function Φ(u) defined in Gerber and Shiu (1998). Φ(u) is given by where e −δTu is the discount factor, ω is the penalty function of surplus before ruin U (T − u ) and deficit at ruin U (T u ). Thus, the ruin probability ψ(u) is a special case of Gerber-Shiu function when δ = 0 and ω = 1.
Assuming that the distribution of the interclaim times (τ k ) k 0 and the claim sizes (X k ) k 0 have rational Laplace transform, the density functions f τ (t), f X (x) satisfy linear ordinary differential equation For the risk models with surplus-dependent premiums, Albrecher et al. (2013) derived a compact integro-differential equation for Φ(u) For a Gerber-Shiu function, the coefficients of ODE are variables (non-constant), and the boundary value problem developed by (Albrecher et al., 2013) is The general solution of this boundary value problem has the form where s i (u), i = 1, . . . , m are m stable solutions (s i (u) → 0 as u → ∞), γ i are constants determined by initial conditions, g(u) = α 0 L X ( d du )ω(u), and Gg(u) is the Green's operator for (2.3) (see Albrecher et al. (2013)).
In next sections we developed above theory to analyse the case when either generic interarrival time or generic claim size has Gamma (Erlang) distribution. We start for more easy case when both, generic claim and generic interarrival time, have exponential distributions.
For a classical compound Poisson process with exponential claims, the following explicit and asymptotic results for ruin probability ψ(u) can be found in Asmussen and Albrecher (2010); Albrecher et al. (2013).
General premium. For a classical compound Poisson process with exponential claims, the ruin probability ψ(u) has the following explicit expression The asymptotic estimate of ruin probability for p(∞) = c is for some functions f and g when lim u→+∞ f (u)/g(u) = 1.
We denote by ψ l,2 (u) and Φ l,2 (u) the ruin probability and Gerber-Shiu function in this case.
3.1. General premium. Based on the technique as in Albrecher et al. (2010Albrecher et al. ( , 2013, the boundary value problem (2.3) becomes For the special case δ = 0 and ω = 1, g(u) = 0, the ODE of the ruin probability ψ l,2 (u) has the form (3.1) where s 21 (u) is a stable solution and γ 21 is a constant to be determined by the initial conditions.
Expanding ODE (3.1) leads to This is a third-order ODE with variable coefficients. Considering the third-order as second-order ODE in h l,2 (u) = ψ ′ l,2 (u), one has In order to perform the asymptotic analysis as in (Fedoryuk, 1993, p. 250), we consider the characteristic equation of (3.2) when p(u) = c. Letρ 1 andρ 2 be solutions of the square equation Moreover, let be solutions of the characteristic equation ρ 2 + q 1 (u)ρ + q 0 (u) = 0, where Further, as in (Fedoryuk, 1993), denote (3.6) ρ (1) Remark 1. Under complimentary to (3.7) assumption 2c λ < 1 µ we haveρ 1,2 > 0 and hence both asymptotic special solutions are unstable. Their difference might still tend to zero but Fedoryuk (1993) theory is not sufficient precise to recover the finer asymptotics in this case.
Observe that indeed in all considered cases ψ l,2 (u) → 0 as u → +∞, that is, we choose the asymptotics of stable solutions.
3.2. Linear premium. Now we perform the asymptotic analysis of the special case of linear premium rate which corresponds to investments of reserves into bonds with interest rate ε > 0. Substituting p(u) = c + εu into ODE (3.2), we have Before we solve this equation and perform the asymptotic analysis we will show how the asymptotics of ψ l,2 can be derived from Theorem 1. In this case, we have Further, the discriminant is Applying Taylor expansion, we can conclude that Additionally, observe that This gives that Using (3.6) we finally derive Thus, for u → ∞, and by (3.9) (3.14) for some constant C 3 . The same asymptotics can be derived by solving (3.13) explicitly. Note that (3.13) is the general confluent equation 13.1.35 in Abramowitz and Stegun (1965, p. 505), which has the form For our ODE (3.13), let the corresponding solutions are where C 21 and C 22 are constants.
By (3.14) this asymptotic behaviour is the same as the one derived using the Theorem 1. Furthermore, one can simplify the above asymptotics by applying the integration-by-parts formula into

4.2.
Linear premium. Using the same method as in the previous case and considering linear premium p(u) = c + εu, one has (4.15) (c + εu)h ′′ l,3 (u) + (2ε + 2µ(c + εu) − λ)h ′ l,3 (u) + (2µε + µ 2 (c + εu) − 2µλ)h l,3 (u) = 0. As in the previous section, before we solve this equation explicitly and then using this solution to perform the asymptotic analysis, we will first show how the asymptotic behaviour of ψ l,3 can be derived from Theorem 2. Note that in this case, we denotẽ Further, we have the discriminant (see Hipp (1998, 2001)). Taking the limit and applying L'Hôpital's rule, the ratio between ψ l,2 (u) and ψ c,2 (u) behaves asymptotically as where C 3 is some constant. Hence ψ l,2 (u) ψc,2(u) tends to zero as u tends to infinity.
This means that as the initial surplus u increases, one has more premium income for risk models with linear premiums, thus the ruin probability ψ l,2 (u) for risk models with linear premiums decays to zero exponentially faster than the ruin probability ψ c,2 (u) for constant premiums risk models. As expected, this means that risk models with linear premiums are less risky than the constant premiums ones.
Thus, as the initial surplus u increases, the ruin probability ψ l,3 (u) for risk models with linear premiums decreases to zero faster than the ruin probability ψ c,3 (u) for constant premiums. Again, this means that risk models with constant premiums are more risky than linear premiums ones, as expected, thus there is gain in terms of solvency when binding premium to reserves.

Conclusion
It is much easier to calculate the ruin probabilities for risk models with constant premiums, and explicit results for constant cases abound in risk theory literature, but the risk models with surplusdependent premiums are more applicable in real life. For these complex cases, we have results in terms of confluent hypergeometric function and modified Bessel function at most, or only asymptotic results, from which one can make inferences.