Risk Process based on General Compound Hawkes Process and its Implementation with Real Data

Transcription

1 Risk Process based on General Compound Hawkes Process and its Implementation with Real Data text Gabriela Zeller University of Calgary Calgary, Alberta, Canada 'Hawks Seminar' Talk Dept. of Math. & Stat. Calgary, Canada August 07, 2018

2 Outline of Presentation The Classical Risk Model Hawkes Process Background RMGCHP: Theoretical Results RMGCHP: Implementation with Real Data Evaluation and Next Steps References

3 The Classical Risk Model This section gives some mathematical background on the classical risk model, such as common assumptions and approaches.

4 The Cramer-Lundberg Model - Background The classical risk model or compound-poisson risk model was introduced in 1903 by Filip Lundberg and has the following form: R(t) = u + ct N t Y i i=1 where u denotes the initial capital, c denotes the (continuous) premium rate and the number of claims in the interval [0, t) is a homogeneous Poisson process N t with rate λ. The claims Y i are i.i.d. positive random variables with distribution function G and mean µ G independent of (N t ).

5

6 The Cramer-Lundberg Model - Computations Quantities of interest that have been extensively studied: Ruin time: τ = inf{t > 0 : R(t) < 0} where inf = Probability of ruin in [0, t): Φ(u, t) = P (τ t R(0) = u) = P ( inf R(s) < 0) Probability of ultimate ruin: 0<s t Φ(u) = lim Φ(u, t) = P (inf R(t) < 0) t t>0 For practical purposes, it is sometimes more convenient to use the survival probability: δ(u) = 1 Φ(u) Severity of ruin: R(τ) Distribution of the time to ruin (given ruin): P (τ < t τ < ) Net Prot Condition: When is the mean income strictly larger than the mean outow? Premium Principle: How should the premium be set against the insurance risk?

7 The Cramer-Lundberg Model - Net Prot Condition The rst interest of an insurance company is to avoid a state where ruin occurs with probability 1. If c < λµ G, ruin is unavoidable, that is Φ(u) = 1. If c = λµ G ruin also occurs with probability 1. (This result requires some deep theory of random walks (e.g. Mikosch (2000)) Thus, the obvious condition for solvency of the company is the net prot condition c > λµ G which implies the premium rate to be c = (1+θ)λµ G where θ is called the safety loading.

8 The Cramer-Lundberg Model - Ruin Probability The following is known for the survival probability (see e.g. Asmussen and Albrecher (2010)): Theorem 1 The survival probability δ(u) is continuous and differentiable everywhere (except the countable set where G is not continuous) and satises the following integro-dierential equation: cδ (u) = λ[δ(u) u 0 δ(u y)dg(y)] (1) For a general distribution G, equation (1) is not analytically solvable.

9 The Cramer-Lundberg Model - Ruin Probability However, for claims following an Exp(γ) distribution, the survival (resp. ruin) probability can be explicitly computed as: δ(u) = 1 λ cγ e(λ c γ)u textandtextφ(u) = λ cγ e(λ c γ)u This convenience is one of the reasons why claims are often assumed to be i.i.d. Exp(γ) distributed for insurance applications (for the light-tailed case).

10 The Cramer-Lundberg Model - Ruin Probability As equation (1) cannot be solved analytically for the general case, this has motivated seeking bounds for the ruin probability. To this end, Lundberg introduced the adjustment coecient R > 0 as the unique positive root of h(r) = 0 where h(r) = λ(m Y (r) 1) cr where M Y (r) = E[e ry 1] is the m.g.f. of (Y i ) which is assumed to be nite for all 0 < r < γ with lim r γ M Y (r) =. The intuition here is that for r R such that M Y (r) < we can show that the process {e rr t h(r)t } t 0 is a martingale.

11 The Cramer-Lundberg Model - Ruin Probability The adjustment coecient is connected to the ruin probability through the following theorem (see e.g. Asmussen and Albrecher (2010)): Theorem 2 (Lundberg Inequality) Φ(u) e Ru, text u 0 where R is the adjustment coecient. For the special case of Exp(γ)-distributed claim sizes, the inequality gives Φ(u) e (λ c γ)u which is related closely to the net prot condition c > λ γ in this case.

12 The Cramer-Lundberg Model - Ruin Probability Using the adjustment coecient, we can furthermore describe the asymptotic behavior of the ruin probability. Theorem 3 Assume that the adjustment coecient R exists and that Then λ c 0 xe Rx (1 G(x))dx < lim u Φ(u)eRu = c λµ G λm Y (R) c

13 The Cramer-Lundberg Model - Diusion Approximation For a general distribution function, ruin probabilities can be estimated using the following diusion approximation. Proposition 1 Let (R t (n) ) be a sequence of Cramer-Lundberg processes with initial capital u, claim arrival intensities λ (n) = nλ, claim size distributions G (n) (x) = G( nx) and premium rates Let µ G = Then c (n) = (1 + c λµ G )λ (n) µ (n) λµ G n G 0 = c + ( n 1)λµ G ydg(y) and assume that µ 2,G = R (n) t d (u + W t ) 0 y 2 dg(y) <.

14 where (W t ) is a (c λµ G, λµ 2,G )-Brownian motion and the convergence is in distribution in the topology of uniform convergence on nite intervals.

15 The Cramer-Lundberg Model - Diusion Approximation Let τ (n) denote the ruin time of (R t (n) ) and τ = inf{t 0 : u + W t < 0} the ruin time of the diusion limit. Proposition 2 Let (R t (n) ) and (W t ) be as above. Then and lim n P [τ (n) t] = P [τ t] lim n P [τ (n) ] = P [τ ]

16 The Cramer-Lundberg Model - Diusion Approximation The idea is to approximate P [τ (1) t] by P [τ t] (nite ruin probability) and P [τ (1) ] by P [τ ] (innite ruin probability). Thus, we use the following result for the Brownian motion: Proposition 3 Let (W t ) be a (m, σ 2 )-Brownian motion with m > 0 and τ = inf{t 0 : u + W t < 0}. Then P [τ < ] = e 2um σ 2 and P [τ t] = 1 Φ( mt + u σ t ) + e 2um σ 2 Φ( mt u σ t )

17 The Cramer-Lundberg Model - Extensions The Cramer-Lundberg risk model is a convenient foundation for classical risk theory, but it does not reect the dependencies of incoming claims in reality. Thus, several extensions have been proposed: The Sparre-Andersen model / renewal risk model introduced in 1957 replaces the Poisson process with a general renewal process N t, thus allowing for claim inter-arrival times with arbitrary distribution functions. However, the lengths of the intervals between subsequent arrivals stay independent. More recently, risk models using a Cox process (with Poisson shot noise) have been studied as they incorporate time-dependent intensity inuenced by exogenously caused jumps ("shocks", environmental factors).

18 The Cramer-Lundberg Model - Extensions However, it has been observed that eects like contagion and clustering in nancial contexts are often caused endogenously. Therefore, Hawkes processes have gained attention due to their ability to reect endogenously caused jumps of the intensity function. Hawkes processes have e.g. been successfully applied to construct stock price models including nancial contagion, to model mid-price changes in high-frequency trading and in order book ow modelling (Bacry (2015) gives a good overview).

19 Hawkes Processes This section gives an introduction to Hawkes processes. First, we give the denition, explain the conditional intensity function and highlight the immigration-birth-representation of a Hawkes process. We then focus on the special case of a Hawkes process with exponentially decaying intensity and give some properties of the number of jumps of such a process over a xed interval. The last part of this section focuses on simulation, parameter estimation and goodness of t testing of a Hawkes model.

20 Hawkes Process: Counting Process The Hawkes process introduced by Hawkes (1971) is a simple point process that can model a sequence of arrivals over time, e.g. trade orders, bank defaults or incoming claims. The counting process N(t) refers to the number of arrivals over time, and the corresponding point process (t 1, t 2,...) refers to the arrival times (thus the "jumps" of N(t)). Consider a counting process (N(t) : t 0) with history (H(t) : t 0) that satises: P(N(t + h) N(t) = m H(t)) = λ (t)h + o(h), m = 1 o(h) m > 1 1 λ (t)h + o(h), m = 0

21 Hawkes Process: Counting Process

22 Hawkes Process: Conditional Intensity Function The Hawkes process has self-exciting property, clustering eect and long memory. This is reected in the conditional intensity function: λ (t) = λ + t 0 µ(t s)dn(s) where λ > 0 is the background intensity and µ > 0 is the excitation function describing how much the intensity is aected by past jumps. Using the observed sequence of arrival times (t 1, t 2,, t k ) up to time t, the conditional intensity can be written as: λ (t) = λ + t i <t µ(t t i )

23 Hawkes Process: Conditional Intensity Function We can see that each new arrival causes the conditional intensity function to jump up instaneously, then decay back to the background intensity until it jumps again at the next arrival. The intensity depends on the whole past history of the process. Platzhalter

24 Hawkes Process: Conditional Intensity Function Of course we can analogously dene a multi-dimensional Hawkes process: Denition 1 A Hawkes process is a counting process N t such that the intensity vector can be written as λ i t = µ i + D j=1 φ ij (t t )dn j t (2) where the quantity µ = {µ i } D i=1 is a vector of exogenous intensities and Φ(t) = {φ ij (t)} D i,j=1 is a matrix kernel which is component-wise positive (φ ij (t) 0 for each 1 i, j D), component-wise causal (if t < 0, φ ij (t) = 0 for each 1 i, j D) and each component φ ij (t) belongs to the space of L 1 -integrable functions.

25 Hawkes Process: Immigration-Birth-Representation A Hawkes process X can be represented as a Poisson cluster process with the following structure: a) Immigrants (cluster centers) are distributed according to a homogeneous Poisson process I with points X i (0, ) and intensity λ > 0. b) Each immigrant (of generation 0) X i generates a cluster C i = C Xi, a random set with the following branching structure: Given generations 0, 1,..., n C i, each Y C i (of generation n) generates a Poisson process on (Y, ) of ospring (of generation n + 1) with intensity function µ( Y ), where µ : (0, ) (0, ] is a non-neg. Borel function called fertility rate. c) Given the immigrants, the centered clusters C i X i = {Y X i : Y C i } for X i I are i.i.d. and independent of I. d) X is the union of all clusters C i. i

26 Hawkes Process: Immigration-Birth-Representation Let 0 < n := µ(s)ds < 1. 0 If an immigrant enters the system at time t i R, they produce ospring at rate µ(t t i ) at future times t > t i. Their ospring (called rst generation) again produces ospring (second generation) and so on - members of all generations are called descendants of the original arrival. Let Z i denote the random number of ospring in the n th generation (where Z 0 = 1 denotes the immigrant). Then E[Z i ] = n i and the expected number of descendants for one immigrant is E[ i=1 Z i ] = i=1 E[Z i ] = i=1 n i = n 1 n, n < 1 n 1

27 Hawkes Process: Immigration-Birth-Representation We can see that the condition n < 1 avoids explosion of the process (where one immigrant would generate innitely many children). Most properties of the Hawkes process rely on this so-called stationarity which we will always assume to hold from now on. Note that for n (0, 1), the branching ratio n can be interpreted as the probability that a random arrival was generated endogenously (a child) as we can look at the ratio of descendants over the whole "family" (descendants + original immigrant): E[ Z i ] i=1 1 + E[ Z i ] i=1 = n 1 n 1 + n 1 n = n

28 Hawkes Process: Immigration-Birth-Representation The immigration-birth representation is interesting for us due to its interpretation in the risk model context: We observe standard claims which arrive according to the points of I and trigger other claims according to the branching structure described before. The fertility rate is mon. decreasing, so the process has a selfexciting structure. This risk process is closely related to the shot-noise Cox model (doubly stochastic Poisson model) studied e.g. in Albrecher (2006), with the main dierence that the other model incorporates time-dependent intensity inuenced by exogenously caused jumps ("shocks", environmental factors) as opposed to endogenously caused clustering in the Hawkes model.

29 Hawkes Process: Compensator Denition 2 (Compensator) For a counting process N( ) the non-decreasing function Λ(t) = t 0 λ (s)ds is called the compensator of the process. Note that M(t) := N(t) Λ(t) is a local H(t)-martingale.

30 Exponential Hawkes Process: Denition We now want to make a concrete choice for the excitation function of the Hawkes process, where we choose the most commonly used exponential decay, thus µ(t) = αe βt. This might seem restrictive (and it is), but until now (almost) all applications of Hawkes processes use this specication as it simplies many theoretical derivations as we will see in the following. Thus, from now on our conditional intensity function will be: λ (t) = λ + α e β(t t i) t i <t Thus, each arrival makes the intensity immediately jump up by α and over time the impact of the arrival decays exponentially at rate β.

31 Exponential Hawkes Process: Denition For the special case of exponentially decaying intensity, given an initial condition λ (t) = λ 0, the conditional intensity process satises the SDE dλ (t) = β(λ λ (t))dt + αdn(t) (3) Also note that the joint process X(t) = (λ (t), N(t)) is a Markov process on the state space D = R + N which allows Da Fonseca et al. (2014) to derive the useful results about properties of the number of jumps of a Hawkes process over a xed interval described in the following.

32 Exponential Hawkes Process: Properties Proposition 4 Given a Hawkes process X(t) = (λ (t), N(t)) with dynamic given by (3), the long-run expected value of the number of jumps during an interval of length τ is given by: lim E[N(t + τ) N(t)] = λ t The variance is given by: 1 α/β τ = lim t E[λ (t)]τ (4) V (τ) = lim E[(N(t + τ) N(t)) 2 ] E[N(t + τ) N(t)] 2 (5) t = λ 1 α/β (τ( α/β )2 + (1 ( 1 α/β )2 ) (1 e τ(β α) ) β α (6)

33 The covariance for two non-overlapping intervals of lengthτ with lag δ > 0 is given by: Cov(τ, δ) = lim t E[(N(t + τ) N(t))(N(t + 2τ + δ) N(t + τ + δ))] (7) E[N(t + τ) N(t)]E[N(t + 2τ + δ) N(t + τ + δ)] (8) = λβα(2β α)(e(α β)τ 1) 2 2(α β) 4 e (α β)δ (9) Note that taking the limit for t (thus putting the process into its long-run stationary regime) to simplify dependence with respect to the initial value λ 0 requires again the stability of the process, which for exponential intensity means αe βs ds = α β < 1. 0

34 Exponential Hawkes Process: Properties Proposition 5 A direct consequence from the last result is the autocorrelation function of the number of jumps over intervals of length τ separated by a time lag of δ: Acf(τ, δ) = e 2βτ (e ατ e βτ ) 2 α(α 2β) 2(α(α 2β)(e (α β)τ 1) + β 2 τ(α β)) e(α β)δ (10) Note that this expression is always positive for α < β (stationarity condition) and it is exponentially decaying with the lag δ.

35 Exponential Hawkes Process: Simulation In order to simulate a Hawkes Process, there are dierent possible methods. We elect to use the modied thinning algorithm introduced in Ogata (1981) and described again in Laub et al. (2015). The original thinning algorithm was used by Lewis et al. (1979) to simulate a non-homogeneous Poisson process with time-dependent rate λ(t) by generating a homogeneous Poisson process with rate M > λ( ) and probabilistically removing points so that the remaining points satisfy the intensity λ(t). An analogous approach that requires updating the upper bound M during the simulation can be used to simulate a Hawkes process.

36 Exponential Hawkes Process: Simulation text

37 Exponential Hawkes Process: Simulation text

38 Exponential Hawkes Process: Parameter Estimation Given a set of arrival times (t 1, t 2,, t k ) assumed to come from a Hawkes process, we would like to generate parameter estimates (ˆλ, ˆα, ˆβ) by the method of maximum likelihood estimation. We start by citing the result from Daley, Vere-Jones (2003): Theorem 4 (Hawkes Process Likelihood) Let N( ) be a regular point process on [0, T ] for some nite T > 0 and let t 1,, t k be a realisation of N( ) over [0, T ]. Then the likelihood L of N( ) is expressible in the form L = ( k i=1 λ (t i ))exp( T 0 λ (s)ds) (11)

39 Exponential Hawkes Process: Parameter Estimation Given the likelihood function from (11), the log-likelihood for the interval [0, t k ] is given as l = k i=1 log(λ (t i )) t k 0 λ (s)ds = k i=1 log(λ (t i )) Λ(t k ) (12) For a Hawkes process with exponential decay, the compensator can be explicitly computed as Λ(t k ) = λt k α β k i=1 [e β(t k t i ) 1] (13)

40 Exponential Hawkes Process: Parameter Estimation In order to make the computation feasible, the term A(i) = 0 i = 1 i 1 j=1 e β(t i t j ) = e β(t i t i 1 ) (1 + A(i 1)) i {2,, k} is introduced, such that the log-likelihood (12) of a Hawkes process with exponentially decaying intensity is given by l = k i=1 log(λ + αa(i)) λt k + α β k i=1 [e β(t k t i ) 1] (14)

41 Exponential Hawkes Process: Parameter Estimation In general, the maximum likelihood estimation will be very effective and its consistency, asymptotic normality and eciency were proved in Ogata (1978). However, for real datasets there are several signicant challenges such as bias for small sample sizes, high number of local optima and O(k) complexity for large sample sizes which made Filiminov et al. (2013) state that "Our overall conclusion is that calibrating the Hawkes process is akin to an excursion within a mineeld that requires expert and careful testing before any conclusive step can be taken."

42 Exponential Hawkes Process: Goodness of Model Fit After estimating parameters, the next important step is assessing the goodness of t of the Hawkes process model for real data. An essential result for this is the following theorem from e.g. Brown et al. (2002). Theorem 5 (Random Time Change Theorem) Let {t 1, t 2,, t k } be a realisation over time [0, T ] from a point process with conditional intensity function λ ( ). If λ ( ) is positive over [0, T ] and Λ(T ) < a.s. then the transformed points {t 1,, t k } = {Λ(t 1 ),, Λ(t k )} form a Poisson process with unit rate. As we know the closed form of the compensator (13), we can test the quality of the parameter estimation by transforming the original timepoints and performing standard tness tests for a unit rate Poisson process on the transformed datapoints.

43 Exponential Hawkes Process: Goodness of Model Fit As suggested in Laub et al. (2015), we use the following steps: QQ-plot of tranformed interarrival times{t 1, t 2 t 1, } against Exp(1)-distribution Independence plot of the points (U i+1, U i ), where U i = F (t i t i 1 ) = 1 e (t i t i 1 ) Checking for autocorrelation in the sequence of transformed interarrival times

44 Risk Model with Hawkes Process and RMGCHP Now we combine the rst two sections by studying a risk model where claims arrive according to a Hawkes process. First, we mention some results obtained in this area by past work. We then introduce the Risk Model with General Compound Hawkes Processes (RMCGHP) suggested by Swishchuk (2017) and the theoretical results obtained for it (Law of Large Numbers and Functional Central Limit Theorem).

45 Risk Model with Hawkes Process The rst work to consider a risk model with Hawkes claims arrivals was Stabile et al. (2010), who derive the asymptotic behavior of innite and nite horizon ruin probabilities and asymptotically ecient simulation laws using that compound Hawkes process fulls a large deviation principle and assuming light-tailed claims. Their work was extended by Zhu (2013) who considered (subexponential) heavy tailed claims.

46 Risk Model with Hawkes Process Dassios and Zhao (2012) consider a risk process with the arrival of claims modelled by a dynamic contagion process, generalising the Hawkes process and the Cox process with shot noise intensity and thus including both self-excited and externally excited jumps. They derive generalisations of the Cramer-Lundberg inequality, Lundbergs equation, some asymptotics as well as bounds for the probability of ruin with special attention on the case of exponential jumps. Dassios and Jang (2012) study a bivariate shot noise self-exciting process for insurance, including a constant rate of exponential decay that could be interpreted as the time value of money. They derive theoretical distributional properties and use numerical examples to show that this point process could be used for the modelling of discounted aggregate losses from catastrophic events.

47 Risk Model with Hawkes Process Cheng and Seol (2018) derive diusion approximations and thus expressions for the ruin probabilities of risk models with Hawkes claims arrivals, providing numerical examples for exponential and Gamma-distributed jumps. They construct the diusion approximation analogously to the classical case and nd that the diusion limit is a Gaussian process that can be decomposed into a centered Gaussian process and an independent Brownian motion. Contrary to the classical case, the variance function of the diffusion limit is nonlinear in t in general, and can be computed explicitly for a Hawkes process with exponential decay.

48 Risk Model Based On General Compound Hawkes Process As a generalisation of the classical risk model, Swishchuk (2017) proposes a risk model with general compound Hawkes process (RMGCHP): R(t) = u + ct N t k=1 a(x k ) (15) where u denotes the initial capital, c denotes the (continuous) premium rate and the number of claims in the interval [0, t) is a Hawkes process N t. The claim sizes X k follow a continuous-time Markov chain on state space X = {1,..., n} independent of N t and a(x) is a continuous and bounded function on X. Special cases of this RMGCHP would be a(x k ) := X k following a Markov Chain and a(x k ) := X k i.i.d.

49 RMGCHP: Theoretical Results The rst important result of Swishchuk (2017) is a Law of Large Numbers for the RMGCHP: Theorem 6 (LLN) Let R(t) be the risk model dened in (15), and let X k be a Markov Chain with state space X and stationary probabilities πn. We suppose that 0 < ˆµ = µ(s)ds < 1. Then where a = k X a(k)π k R(t) lim t t = c a λ 1 ˆµ 0 (16)

50 RMGCHP: Theoretical Results From this Law of Large Numbers follow the net prot condition and premium principle: Corollary 1 (Net Prot Condition) c > a λ 1 ˆµ (17) Corollary 2 (Premium Principle) where θ is the safety loading. c = (1 + θ)a λ 1 ˆµ (18)

51 RMGCHP: Theoretical Results The second important result in Swishchuk (2017) is the following Functional Central Limit Theorem: Theorem 7 (FCLT) Let R(t) be the risk model dened in (15), and X k be an ergodic Markov Chain with stationary probabilities πn. We suppose that 0 < ˆµ = µ(s)ds < 1 and Then: 0 R(t) (ct a N(t)) D lim = σφ(0, 1) t t (or in Skorokhod topology (see Skorokhod (1965))): 0 sµ(s)ds <. R(nt) (cnt a N(nt)) D lim = σw (t) n (19) n

52 where Φ(, ) is the std. Normal cdf and W(t) is a std. Wiener process. σ := σ λ/(1 ˆµ), (σ ) 2 := a := i X ν i := b(i) 2 + 2b(i) i X π i a(i), b(i) := a a(i) j X j X π i ν(i) (g(j) g(i)) 2 P (i, j) (g(j) g(i))p (i, j) g := (P + Π I) 1 (b(1),..., b(n)) T where P is the transition matrix for X k and Π is the matrix of stationary probabilities of P.

53 RMGCHP: Theoretical Results The FCLT allows us to approximate the risk process R(t) by the jump-diusion-process D(t): R(t) u + ct N(t)a + σw (t) := u + D(t) where a and σ are dened as above, N(t) is a Hawkes process and W (t) is a standard Wiener process. The rate of approximation is given by E R(t) (ct a N(t)) σw (t) 1 t C(c, a, σ, λ, ˆµ) or E R(tn) (cnt a N(nt)) σw (t) 1 n C(c, a, σ, λ, ˆµ, T ) from Swishchuk (2000).

54 RMGCHP: Theoretical Results We use the diusion approximation to calculate the ruin probability in a nite time interval and the ultimate ruin probability. Theorem 8 (Finite horizon ruin probability) Ψ(u, τ) = Φ( u + (c a λ/(1 ˆµ))τ σ τ + e 2(c a λ/(1 ˆµ)) σ 2 u u (c a λ/(1 ˆµ))τ Φ( σ ) τ ) Theorem 9 (Ultimate ruin probability) Ψ(u) = e 2(c a λ/(1 ˆµ)) σ 2 u

55 Implementation of RMGCHP with Empirical Data We would now like to implement the results for RMGCHP with real data provided by ERGO Group. We rst explain the dataset and data preparation steps. We then show that a Poisson model is not suitable for this data, thus we estimate parameters for a Hawkes model and test the goodness of t of the obtained model with respect to real claim arrival times. Based on the obtained Hawkes model, we use the results from Swishchuk (2017) to calculate premium rates and ruin probabilities using the diusion approximation. In order to test the appropriateness of the approximation, we then compare the standard deviation of the empirical process (for large timescales) with the theoretical diusion coecients.

56 Empirical Data: Description ERGO Group has kindly sent ve very comprehensive data sets, one for each reporting year 2010 to These include claims that were reported to the company in the respective year, although the claim might have occurred in an earlier year and might cause payments from the reporting year on into other years in the future. The data sets are about claims from various kinds of 'Rechtsschutzversicherung' (insurance covering claims from legal disputes, i.e. lawyer fees).

57 Empirical Data: Description Each data set consists of the following columns: VNR: Policy Number (e.g. one client) SCHDLNR: Running number of claim per policy (a client can have several claims over the years) SCHDDAT: Date of claim occurrence (this can be earlier than the reporting year) ZAHLUNG: Claim payment (one claim can result in several payments over time, this is the main point of interest) ZHLGDATUM: Date of the claim payment (temporal structure of this is main point of interest) LEISTUNGSART: Type of insurance (there are many classes of legal expenses insurance, in total 44 classes in this dataset) STATUS: Has the case been closed or is it still open? MELDJAHR: Reporting Year

58 Empirical Data: Description Example of the dataset from the reporting year 2010: text The rst three rows belong to the same policy and claim occurrence (on ). The claim was reported in 2010 and has led to three payments by the insurer on future dates (06.05., and ) with dierent claim sizes (745.78, and 4.43). The case has been closed.

59 Empirical Data: Preparation For a selected subset we extract the columns ZAHLUNG (claim payment) and ZHLGDATUM (claim payment date) and modify ZHLGDATUM as follows: For the reporting year 2010, the starting point 0 is chosen as and the time scale is in days. Each date is thus assigned a number (days from the start), e.g or For tting data from the reporting year 2011 only, the date is chosen as 0 and so on. For tting data from all reporting years simultaneously, the starting point is and numbers are assigned continuously, i.e

60 Empirical Data: Preparation On some days, there is more than one occurrence which is impossible for a simple point process. In this case, occurrences are distributed uniformly (non-random) over the day, e.g. if there are 3 occurrences on day 15, they are assigned 15, and This is questionable, but there is no precedent in literature for insurance data and distributing occurrences randomly uniformly over the day (as has been done over milliseconds for nancial data) leads to very unstable parameter estimations over each run. For the following demonstration, we choose the dataset of claims of type "Firmen-Arbeitsschutz" which have been reported in the years 2010 to 2014 and which have occurred exactly three years before their reporting (thus between 2007 and 2011). This dataset has 1205 events over a period of T = 2882 days.

61 Empirical Data: Check of Poisson Assumption First, we plot the number of payment occurrences per week (7 days) over the whole time period and inspect if clustering can be seen. text

62 Empirical Data: Check of Poisson Assumption To justify the use of a Hawkes process, we next plot the interarrival times of claim occurrences against an exponential distribution in a QQ-plot. If the data followed a memoryless Poisson distribution, we should see a good t. At this point, we can conclude that a classical risk model would not be appropriate.

63 Empirical Data: Parameter Estimation We t parameters ˆλ, ˆα, ˆβ using Maximum Likelihood Optimization where we constrain the parameters by(0, 0, 0) and (500, 500, 500) and vary starting values for each parameter between 0 and 10. This seems reasonable as ˆλ for a Hawkes process should be smaller than the "Poisson" rate of = events per time step and ˆα and ˆβ should not be huge either due to the rather low frequency of events per time step. Parameter estimates are stable within this range as long as the starting values of λ and β are not both much higher than the one for α in which case ˆα = 0 which would not be a Hawkes process. The estimates for this specic dataset are ˆλ = , ˆα = , ˆβ = We observe < 1 indicating stability of the process, although ˆα ˆαˆβ and ˆβ are quite close together.

64 Empirical Data: Goodness of Fit As a rst goodness of t test, we transform arrival times using the closed-form compensator with the estimated parameters and plot their interarrival times against an Exp(1)-distribution. Note: In literature considering Hawkes process tting to empirical data, this criterion is never shown to test the goodness of t (as it is generally dicult to get a good t here).

65 Empirical Data: Goodness of Fit To test independence of tranformed interarrival times, we plot the points (U k, U k+1 ) as described above. Ideally, we should see uniformly scattered points here. Note: Again, this criterion is generally dicult to meet for empirical data.

66 Empirical Data: Goodness of Fit As described above, it is generally dicult to meet the standard goodness of t criteria for empirical data. For most empirical datasets, the goodness of model t as described above per se is mediocre. Realistically, we should be doubtful whether the data follows an exponential Hawkes process (as this is a restrictive assumption). However, even if the data does not strictly follow a Hawkes process with exponential intensity, it might still be possible to describe the characteristics of the data well with a Hawkes process!

67 Empirical Data: Goodness of Fit To this end, we compare the expected value and the autocorrelation of the number of jumps on an interval for the empirical data with the theoretical values derived by Da Fonseca et al. (2014). We nd that for all datasets the Hawkes process estimates the number of jumps over any interval very well. For some datasets, although the t from the QQ-plot above might not be very good, the Hawkes process almost perfectly matches the autocorrelation for intervals of 7 (14,28) days and time lags ranging from 7 to 180 (240) days. Note that a Poisson process would assume the autocorrelation to be 0 for all lags which is clearly not the case for empirical data.

68 Empirical Data: Goodness of Fit For our dataset, we compare the theoretical autocorrelation of the number of jumps of a Hawkes process with ˆλ = , ˆα = , ˆβ = (line) with the autocorrelation of the number of jumps over the same intervals and lags for the empirical arrival times (points). Interval lengths are 28 days and lags range from 7 to 180 by steps of 7.

69 Empirical Data: Goodness of Fit We observe that the empirical autocorrelation is decreasing with the time lag (as it should be for a Hawkes process) but unfortunately the decrease is not exponential. The slow decay might indicate a Hawkes process with power law decay might be more suitable. However, in past literature, many authors state that although empirical evidence generally favours power law decay, they choose to work with exponential decay as it is analytically much more convenient. As our work is the rst one with empirical insurance data, using exponential decay could thus be justied. This slow autocorrelation decay however favours β α 0 which distorts the ratio α β away from the empirical branching ratio ('primary' claims/immigrants vs. 'secondary' claims/children).

70 Empirical Data: Risk Process After tting a Hawkes process to the empirical claim times, we would like to simulate the risk process and implement the results from Swishchuk (2017). Again, the RMGCHP is dened in (15) as R(t) = u + ct N t k=1 a(x k ) where u denotes the initial capital, c denotes the (continuous) premium rate and the number of claims in the interval [0, t) is a Hawkes process N t. The claim sizes X k follow a continuous-time Markov chain on state space X = {1,..., n} independent of (N t ) and a(x) is a continuous and bounded function on X. Special cases of this RMGCHP would be a(x k ) := X k following a Markov Chain and a(x k ) := X k i.i.d.

71 Empirical Data: Risk Process We do all the following computations for the case of i.i.d. claim sizes with two states as well as claims following a Markov Chain with 2,3,4 and 10 states. In order to choose the function values a(x k ), we follow the quantile-based approach used in Swishchuk et al. (2017) described below. For a Markov Chain with n states, we divide the empirical claim sizes along n quantiles and assign to each 'section' the conditional mean claim size. In order to obtain the transition matrix, we count empirical occurrence frequencies of transitions from one claim size to another.

72 Empirical Data: Risk Process For the example dataset, e.g. for four states, we would obtain the conditional mean claimsizes a = (a(1), a(2), a(3), a(4)) = ( , , , ) and transition matrix

73 Empirical Data: Risk Process In the next step, we can nd the stationary distribution as π = (π1, π 2, π 3, π 4 ) = ( , , , 0.25) Then we can nd the value of a required for the LLN (16) and FCLT (19): a = k X a(x k )π k = = Note that this value is very close to the mean claimsize of as should be expected for our choice of a(x k ) and the stationary distribution above.

74 Empirical Data: Risk Process As we do not have information on the initial capital u or on the premium rate c, we set the initial capital for our dataset as u = 100 and calculate the premium rate using the expected value principle in (18) with a safety loading θ = 0.1: c = (1 + θ)a λ 1 ˆµ =

75 Empirical Data: Risk Process Given u and c, we can now plot the empirical risk process using the actual claimsizes and claim occurrence times: Note that the nal capital at time T for the empirical process is (if ruin did not occur before time T here).

76 Empirical Data: Risk Process Analogously, we can simulate risk processes using a Hawkes process with the estimated parameters ˆλ = , ˆα = , ˆβ = as arrival times and a simulated Markov Chain with transition matrix P and claim sizes a(x k ) for the dierent states:

77 Empirical Data: Risk Process In order to evaluate how well the simulated risk process ts the empirical one, we could use the following criteria adapted from Zhang (2016): First, we look at how well the simulated process R(t) ˆ matches the empirical one in terms of uctuations: Ŝ(L) = 1 L L i=1 max( ˆR i (t)) min( ˆR i (t)) max(r(t)) min(r(t)) where L is the number of simulated paths. Furthermore, we consider how far the nal capital of simulated paths is from the empirical one by considering: ˆF (L) = 1 L L i=1 ˆR i (T ) R(T )

78 Empirical Data: Risk Process We compare the results for a "classical" Poisson risk model with i.i.d. claim sizes and three Hawkes models with Hawkes process arrivals and i.i.d. claims (two states) and Markov Chain claims (2,3,4,10 states) (as the results are very similar for all Hawkes models, we only print 2 and 10 state case here). We use L = 1000 simulations. We can see that a Poisson model would underestimate uctuation whereas a Hawkes model (independent of the claim size modelling) overestimates it. Furthermore, a Hawkes model unfortunately systematically overestimates the nal capital (fc).

79 Empirical Data: Risk Process Model Ŝ(N) ˆF (N) Mean (fc) Std. (fc) Poisson Hawkes (i.i.d.) Hawkes ( state MC) Hawkes (10-state MC)

80 Empirical Data: Risk Process We would now like to implement the results from the FCLT and diusion approximation (19) derived in Swishchuk (2017). To this end, we rst compute the values of a, σ and the diusion coecient σ = σ ( λ ) for MC claims with dierent numbers 1 α/β of states. Model a σ σ Hawkes (i.i.d.) Hawkes (2 states MC) Hawkes (3 states MC) Hawkes (4 states MC) Hawkes (10 states MC) Note that a stays constant (which makes sense considering our choice of a(x k )), but σ and accordingly σ increase with the number of states.

81 Empirical Data: Ruin Probabilities Given the diusion approximation from Swishchuk (2017) with diusion coecients σ calculated before, we can use (8) to calculate ruin probabilities over intervals of length T. We compare the obtained numbers with ruin probabilities obtained from L = 1000 simulations of the risk process until time T. Model Theoretical RP Simulated RP Hawkes (i.i.d.) Hawkes (2-state MC) Hawkes (3-state MC) Hawkes (4-state MC) Hawkes (10-state MC)

82 Empirical Data: Error Estimation In order to understand the discrepancies between theoretical and empirical ruin probabilities, we would like to assess the appropriateness of the diusion approximation. We proceed as suggested in Swishchuk et al. (2017). Given the FCLT R(nt) (cnt a N(nt)) lim n n D = σw (t) we would like to compare the standard deviation of the RHS multiplied by n, that is n tσ λ to the counterpart on 1 α/β the LHS, that is the standard deviation of the process R(nt) (cnt a N(nt)) = u N(nt) k=1 (a(x k ) a ) (20)

83 Empirical Data: Error Estimation For the empirical dataset, we choose t to be our original time scale of one day and rst let n run from 7 to 1435 by steps of 7. At each step nt, we compute the value of the process (R(int) (cint a N(int))) (R((i 1)nt) (c(i 1)nt a N((i 1)nt))), thus we consider intervals of e.g. one week in the rst step. We then compare the standard deviation of these values to the standard deviation theoretically obtained on the RHS of (19) multiplied by n. Note that this approximation should be naturally only accurate for large n, however due to our relatively short time frame of 2882 days, for large n the LHS of (19) is based on only few observations.

84 Empirical Data: Error Estimation The following plot shows the empirical standard deviation (points) for dierent models against the theoretical standard deviation (lines). The empirical standard deviation is only plotted for a model with i.i.d. claims (two sizes), as the results look very similar for all models on this scale.

85 Empirical Data: Error Estimation We have to keep in mind that empirical standard deviations for large values of nt are based on very few samples. If we look at the standard deviations for a sequence ofn from 1 to 35 (ve weeks) in steps of 1 day, we obtain the following picture where now all values are based on at least 80 observations. The rst plot shows the result for a Hawkes model with i.i.d. claims (two sizes).

86 Empirical Data: Error Estimation We now compare the results for Hawkes models with Markov Chain claims with dierent numbers of states.

87 Evaluation and Next Steps In the following, I would like to summarize some concerns about the results obtained so far and possible next steps: 1. Parameter Estimation for Hawkes process leads to good model ts (according to QQ-plot of transformed interarrival times and independence plot) only for a minority of datasets. Note: Most applications of Hawkes processes to empirical data do not show the Hawkes process t per se, but nd other criteria for model t (i.e. autocorrelation t or signature plot for nancial data). We could thus use other criteria, e.g. good ruin probability approximations or error estimations for the diusion approximation.

88 Evaluation and Next Steps 2. Data Description and Autocorrelation plot suggest that a Hawkes process (with exponential decay) is not perfectly suitable. As we can see from the empirical datasets, the response to an initial claim arrival is not instantaneous, but there is usually a time period between related claim payments (weeks or months on a total timescale of seven years, thus we cannot make it close to instantaneous by compressing the timescale). Considering our work is the rst one with Hawkes processes for empirical insurance data, its use could probably still be justied, but we have to consider this when evaluating risk model ts.

89 Next Steps: Power Law Kernel One possibility is tting a Hawkes process with power-law kernel instead of an exponential kernel. In this case, the conditional intensity function is λ (t) = λ + t i <t k (t t i + c) 1+η where k, c, η > 0. In this case, interpretation of the parameters is not so straightforward, but roughly one could say that k corresponds to α describing the upward jump of the intensity caused by an arrival (the magnitude of its inuence), η corresponds to β determining how longlasting the impact of the arrival is and c describes a temporal shift to keep the intensity bounded when (t t i ) is close to 0.

90 Next Steps: Power Law Kernel For a power law kernel we nd that the stationarity condition is kc η and the compensator is given as Λ(t) = t 0 η λ (s)ds = λt + k η < 1 t j <t (c η (t t j + c) η )

91 Next Steps: Power Law Kernel Thus, we can deduce the loglikelihood function analogously as for the exponential case in Laub et al. (2015) and given in (14) and get: l = n i=1 i 1 log(λ+k i t j +c) j=1(t (1+η) ) λt + k η n i=1 ((T t i +c) η c η ) where (t 1,..., t n ) are the observed arrivals over the interval [0, T ]. Unfortunately, in this case there is no easy way to avoid the double summation in the rst part of the loglikelihood function which leads the optimization to be very slow for a large number of arrivals. (In the exponential case, we used a recursive computation derived by Ozaki (1979)).

92 Next Steps: Power Law Kernel One startling observation so far for constrained MLE for a power law kernel is that for simulated data, even given the correct parameters as starting values, the estimated parameters (ˆλ, ˆk, ĉ, ˆη) are rather far away from the real ones, but the conditional intensity function over time looks quite similar. We simulate a Hawkes process with power law kernel and parameters (λ, k, c, η) = (0.5, 1, 1, 2) over an interval of [0, 100]. Parameters are estimated as (ˆλ, ˆk, ĉ, ˆη) = (0.3959, , , ) for an upper bound of 500 for each parameter. We plot the conditional intensity function for both sets of parameters below.

93 Next Steps: Power Law Kernel text

94 Next Steps: Power Law Kernel For the empirical dataset from above, we calculate parameter estimates (ˆλ, ˆk, ĉ, ˆη) using the lower bound (0, 0, 0, 0) and varying the upper bound for each parameter until We nd that ˆλ, ĉ and ˆη are mostly independent of the starting values, but ˆk is always estimated as the upper bound (due to the long computation time, a large number of runs with dierent values was not feasible). We nevertheless plot the intensities with bounds 500 and 5000 and compare them with the intensity from the exponential kernel estimated before. We nd that on the overall time period [0, 2882], the intensities look quite similar, however, if we zoom in on the period of [0, 200], we observe that a power law kernel would favor a faster decay than an exponential kernel (which would contradict our intention of using it). This might of course be caused by the problems when estimating ˆk and could be studied further.

95 Next Steps: Power Law Kernel text

96 Next Steps: Power Law Kernel text

97 Evaluation and Next Steps 3. Risk Model Computations and Ruin Probabilities are inaccurate when comparing theoretical formulas and simulations. This holds for all datasets and even for simulated data when using the correct parameters as estimations. A possible reason for this might be that ˆα ˆβ, so the Hawkes process is close to unstable and thus outcomes are very "volatile" over simulations.

98 Evaluation and Next Steps 4. Claim Size Modelling with a Markov Chain is not consistent with empirical process. The way the empirical insurance claim portfolio is constructed (various claims with more than one payment are bundled together on the same timescale), claim sizes that directly follow each other in the overall portfolio are not necessarily related as they generally don't come from the same claim process. Thus, the use of a Markov Chain that connects claim sizes is not really justiable. One possibility would be to use i.i.d. exponentially distributed claims (as is commonly done in insurance literature).

99 Next Steps: Claim Sizes as i.i.d. Exp(γ) The risk model would have the following form: R(t) = u + ct N t Y i i=1 where u is the initial capital, c is the premium rate and the number of claims in the interval [0, t) is a Hawkes process N t. The claims Y i are i.i.d. Exp(γ)-distributed random variables with mean µ G = 1 γ independent of (N t).

100 Next Steps: Claim Sizes as i.i.d. Exp(γ) In order to check whether this approach would be suitable for our empirical data, we rst plot the empirical c.d.f. of claimsizes against an Exp(γ) distribution where we choose γ = 1 mean(claimsizes).

101 Next Steps: Claim Sizes as i.i.d. Exp(γ) As two further qualitative tests we could again use a QQ-plot against the Exponential distribution and the probability integral transform which should ideally lead to i.i.d. U nif[0, 1] variables (independence plot). text text Both qualitative test lead us to believe that modelling claimsizes as i.i.d. exponentially distributed would be a suitable approach for this dataset.

102 Next Steps: Claim Sizes as i.i.d. Exp(γ) If we compute the diusion approximation for this model based on the sequence of n = (1,..., 35) days, we get the following result. Note that for this case, the value of σ on the RHS of equation (19) would be σ = sd(x k ) = V ar(x k ) = 1 γ which is simply the mean of the empirical claimsizes.

103 Next Steps: Markov-Modulated Claimsizes For other empirical datasets, the assumption of i.i.d. Exp(γ) claimsizes does not seem to hold. Thus, as a generalization, one could consider studying a risk model where claimsizes are assumed to come from two dierent Exponential distributions, say Exp(γ 1 ) and Exp(γ 2 ), depending on the state of an underlying (unobservable) Markov Chain (with two states). Thus, we would have a process (Z t, Y t ) where (Z t ) is a Markov Chain on the state space S = {1, 2} and (Y t ) are the claimsizes of the risk model. The distribution of an incoming claim at time t would only depend on the state of S t and be independent of other (Y ).

104 Next Steps: Markov-Modulated Claimsizes A general case of such a semi-markov risk model (for Poisson arrivals) was studied in Reinhard (1984) and asympotic non-ruin probabilities were obtained explicitly for the case above (two states and Exponential distributions). More recently Yu and Li (2005) gave non-ruin probabilities for more general claim size distributions. Asmussen (1986) studies risk theory in a Markovian environment, studying a variety of methods for assessing the ruin probabilities, in particular Cramér-Lundberg approximations and diusion approximations with correction terms.

105 Next Steps: Diusion Approximation The very recent work of Cheng and Seol (2018) also studies a diusion approximation for a risk model with Hawkes claims arrivals and exponentially distributed claimsizes. They construct the diusion approximation analogously as described for the classical risk model in Chapter 1 and nd that the limit is a Gaussian process with non-linear variance function (as opposed to the diffusion towards a standard Brownian motion for the Poisson case).

106 Next Steps: Diusion Approximation They start with the risk model R(t) = u + ct N t Y i i=1 where (Y i ) are i.i.d. claims with E[Y 1 ] = m 1 < and E[Y1 2] = m 2 <, independent of the claims arrival process (N t ) which is assumed to follow a stationary Hawkes process with intensity λ (t) = λ + µ(t t i ). Let ˆµ = 0 t i <t µ(t)dt <.

107 Next Steps: Diusion Approximation Similarly to the standard diusion approximation in the "classical" case, they construct a sequence of processes (R λ ) (where λ is the background rate of the Hawkes process) as R λ (t) = u + c λ t N λ t i=1 1 λ Y i where c λ = λm1 1 ˆµ + k for a constant k > 0 and N λ t describes the number of arrivals of a Hawkes process with background rate λ. They study the limit for λ.

108 Next Steps: Diusion Approximation Intuitively, we go from a process with few, big jumps to a process with smaller jumps at a higher rate. To illustrate this, we choose a Hawkes process with exponential intensity, i.e. λ (t) = λ + t i <t αe β(t t i) and ˆµ = α β. We assume (Y i ) to be i.i.d.exp(γ)-distributed with mean m 1 = 1 γ. We dene the sequence of risk processes R n (t) = u + c (n) t N (n) t i=1 Y (n) i where N t (n) is the number of arrivals of a Hawkes process with background intensity nλ and (Y (n) i ) are i.i.d. Exp( nγ)-distributed, i.e. E[Y (n) i ] =: m (n) 1 = m1 n.

109 Next Steps: Diusion Approximation The premium rate is set as c (n) = m 1 1 α/β ((1+θ)λ+ n 1), where θ is again the safety loading from the expected value principle (note that for n = 1 we obtain the premium rate we had used before for the "original" process). We then let n.

110 Next Steps: Diusion Approximation The following plot shows realisations of the risk process for different choices of n for a Hawkes process with initial parameters (λ, α, β) = (1, 2, 5) and initial claimsizes according to an exponential distribution with γ = 1/4. The initial capital is set as u = 10 and the safety loading is θ = 0.1.