Modelling the risk process

Modelling the risk process Krzysztof Burnecki Hugo Steinhaus Center Wroc law University of Technology www.im.pwr.wroc.pl/ hugo

Modelling the risk process 1 Risk process If (Ω, F, P) is a probability space carrying: 1. a point process {N t } t 0, i.e. an integer valued stochastic process with N 0 = 0 a.s., N t < for each t < and nondecreasing realizations, and 2. an independent sequence {X k } k=1 variables, of positive i.i.d. random then the risk process {R t } t 0 is given by: R t = u + c(t) N t i=1 X i. (1)

Modelling the risk process 2 Risk process cont. where: R t = u + c(t) N t i=1 X i, {N t } t 0 is the claim arrival point process, {X k } k=1 is an independent claim sequence of positive i.i.d. random variables with common mean µ, u is a nonnegative constant representing the initial capital of the company, c(t) is the premium function.

Modelling the risk process 3 Simulation of the risk process The simulation of the risk process R t or the aggregated claim process { N t i=1 X i} reduces to modeling: the claim arrival point process {N t }, the claim size sequence {X k }, Both processes are assumed to be independent, hence can be simulated independently of each other.

Modelling the risk process 4 Generating continuous random variables Problem: Generate sample of a random variable X with a given density f. (The sample is called a random variate) The simulation of the claim sizes can be done by: inverse transform method, convolution method, composition method, acceptance-rejection method.

Modelling the risk process 5 Inverse transform method Assumption: We can generate U, i.e., uniform (0, 1) random variable. Consider a continuous r.v. having distribution function F. Theorem: For any continuous distribution F the r.v. X defined by X = F 1 (U) has distribution F, where F 1 (u) = inf x : F (x) = u. Thus, the algorithm is: Step 1: Generate a uniform random variable U. Step 2: Set X = F 1 (U).

Modelling the risk process 6 Inverse transform method. Example Exponential r.v.: F (x) = 1 exp( βx). If 1 exp( βx) = u, then x = 1 β log(1 u). X = 1 β log(1 U). Algorithm: Step 1: Generate a uniform random variable U. Step 2: Set X = 1 β log(u).

Modelling the risk process 7 Rejection Method Suppose we can simulate a r.v. with density function g(x). We would like to simulate a r.v. with density function f(y). Let c be a constant such that f(y)/g(y) c, for all y. Then the rejection algorithm is: Step 1: Generate Y having density g and generate U. Step 2: If U f(y) cg(y), then X = Y ; else go to step 1. Theorem. The r.v. X generated by the rejection method has density function f. Moreover, the number of iterations needed to obtain X is a geometric r.v. with mean c.

Modelling the risk process 8 Convolution method Suppose X is a sum of independent random variables Z 1, Z 2,... Z m, i.e. X = Z 1 + Z 2 +... Z m, where Z i F i and are all independent. Algorithm: Step 1: Generate m random numbers U 1, U 2,... U m. Step 2: Inverse transform method: Z i = F 1 (U i ). Step 3: Set X = m i=1 Z i.

Modelling the risk process 9 Convolution method. Example Generate a sample from Erlang(β; m) distribution. Algorithm: Step 1: Generate m random numbers U 1, U 2,... U m. Step 2: Inverse transform method: Z i = 1 β log(u i). Step 3: Set X = m i=1 Z i.

Modelling the risk process 10 Composition method Suppose that either the distribution F X or the probability density f X can be represented either of the following two forms: (1) F X (x) = p 1 F Y1 (x) + p 2 F Y2 (x) +... p m F Ym (x) (2) f X (x) = p 1 f Y1 (x) + p 2 f Y2 (x) +... p m f Ym (x) where p 1,..., p m are non-negative and sum to one (so that they form a probability mass function).

Modelling the risk process 11 Composition method cont. Then, assuming that the Y i s are relatively easily to generate, we can generate X as follows: Step 1: Generate a discrete random variable I on {1,..., m}, where P (I = j) = p j for 1 j m. Step 2: Generate Y I from F YI (or f YI ). Step 3: Return X = Y I.

Modelling the risk process 12 Simulation of the claim arrival point process {N t } is simulated either via: the arrival times {T i }, i.e. moments when the ith claim occurs, or the inter-arrival times (or waiting times) W i = T i T i 1, i.e. the time periods between successive claims. The prominent scenarios for {N t }, are given by: the homogeneous Poisson process (HPP), the non-homogeneous Poisson process (NHPP), the mixed Poisson process, the Cox process (or doubly stochastic Poisson process), the renewal process.

Modelling the risk process 13 Homogeneous Poisson process A continuous-time stochastic process {N t : t 0} is a (homogeneous) Poisson process with intensity (or rate) λ > 0 if: 1. {N t } is a point process, and 2. the times between events are independent and identically distributed with an exponential(λ) distribution, i.e. exponential with mean 1/λ.

Modelling the risk process 14 Simulation of the HPP Successive arrival times T 1, T 2,..., T n of the (homogeneous) Poisson process can be generated by the following algorithm: Step 1: set T 0 = 0 Step 2: for i = 1, 2,..., n do Step 2a: generate an exponential random variable E with intensity λ Step 2b: set T i = T i 1 + E

Modelling the risk process 15 Simulation of the HPP cont. Given that N(t) = n, the n occurrence times T 1, T 2,..., T n have the same distributions at the order statistics corresponding to n i.i.d. random variables uniformly distributed on the interval (0, t]. Hence the arrival times T 1, T 2,..., T n of the HPP on the interval (0, t] can be generated as follows: Step 1: Generate a Poisson random variable N with intensity λ. Let N = n. Step 2: Generate n random variables U i distributed uniformly on (0, 1), i.e. U i U(0, 1), i = 1, 2,..., n. Step 3: (T 1, T 2,..., T n ) = t sort{u 1, U 2,..., U n }.

Modelling the risk process 16 Simulation of the HPP cont. Since EN t = λt, it is natural to define the premium function as: c(t) = ct = (1 + θ)µλt, where µ = EX k and θ > 0 is the relative safety loading which guarantees survival of the insurance company. With such a choice of the premium function we obtain the classical form of the risk process: R t = u + ct N t i=1 X i.

Modelling the risk process 17 Non-homogeneous Poisson process The non-homogeneous Poisson process (NHPP) can be thought of as a Poisson process with a variable intensity defined by the deterministic intensity (rate) function λ(t). A NHPP can model situations, where claim occurrence epochs are likely to depend on the time of the year or of the week. The increments of a NHPP do not have to be stationary. When λ(t) = λ, the NHPP reduces to the HPP with intensity λ. The simulation of the non-homogeneous Poisson process is slightly more complicated than the homogeneous one.

Modelling the risk process 18 Simulation of the NHPP integration method The increment of a NHPP with rate function λ(t) is distributed as a Poisson random variable with intensity λ = t s λ(u)du. Hence, the distribution function F s of the waiting time W s satisfies: F s (t) = P (W s t) = 1 P (W s > t) = 1 P (N s+t N s = 0) = ( s+t ) ( t ) = 1 exp λ(u)du = 1 exp λ(s + v)dv. s If we can find a formula for the inverse Fs 1 then for each s we can easily generate W s using the inverse transform method. 0

Modelling the risk process 19 Simulation of the NHPP integration method cont. The resulting algorithm can be summarized as follows: Step 1: set T 0 = 0 Step 2: for i = 1, 2,..., n do Step 2a: generate a random variable U distributed uniformly on (0, 1), i.e. U U(0, 1) Step 2b: set T i = T i 1 + Fs 1 (U)

Modelling the risk process 20 Simulation of the NHPP thinning Suppose that there exists a constant λ such that λ(t) λ for all t. Let T 1, T 2,... be the arrival times of a HPP with intensity λ. Accept the ith arrival time with probability λ(ti )/λ, independently of all other arrivals, as part of the thinned process (hence the name of the method). The sequence T 1, T 2,... of the accepted arrival times forms a sequence of the arrival times of a NHPP with rate function λ(t). The algorithm amounts to rejecting (hence the alternative name rejection method) or accepting a particular arrival as part of the thinned process.

Modelling the risk process 21 Simulation of the NHPP thinning cont. The resulting algorithm reads as follows: Step 1: set T 0 = 0 and T = 0 Step 2: for i = 1, 2,..., n do Step 2a: generate an exponential random variable E with intensity λ Step 2b: set T = T + E Step 2c: generate a random variable U U(0, 1) Step 2d: if U > λ(t )/λ then return to step 2a ( reject the arrival time) else set T i = T ( accept the arrival time)

Modelling the risk process 22 Simulation of the NHPP cont. Given that N(t) = n, the n occurence times T 1, T 2,..., T n have the same distributions at the order statistics corresponding to n independent random variables distributed on the interval (0, t], each with the common density function f(v) = λ(v)/ t λ(u)du, v (0, t]. 0 Hence the arrival times T 1, T 2,..., T n of the NHPP on the interval (0, t] can be generated as follows: Step 1: Generate a Poisson random variable N with intensity t λ(u)du. Let N = n. 0 Step 2: Generate n random variables V i, i = 1, 2,... n given by the densityf(v) = λ(v)/ t 0 λ(u)du. Step 3: (T 1, T 2,..., T n ) = sort{v 1, V 2,..., V n }.

Modelling the risk process 23 Simulation of the NHPP cont. Since EN t = t λ(s)ds, it is natural to define the premium function 0 in the non-homogeneous case as: c(t) = (1 + θ)µ t 0 λ(s)ds, where µ = EX k and θ > 0 is the relative safety loading. Then the risk process takes the form: R t = u + (1 + θ)µ t 0 λ(s)ds N t i=1 X i.

Modelling the risk process 24 Simulation of the NHPP cont. a = 1, b = 0.01 (blue line), b = 0.1 (red), b = 1 (green) Linear intensity (a+b*t) Seasonal intensity (a+b*sin(2*pi*t)) N(t) 0 10 20 30 40 50 N(t) 0 10 20 30 0 5 10 t 0 5 10 t STFrisk01.xpl STFrisk02.xpl

Modelling the risk process 25 Mixed Poisson process In many situations the portfolio of an insurance company is diversified in the sense that the risks associated with different groups of policy holders are significantly different. For example, in motor insurance we might want to make a difference between male and female drivers or between drivers of different age. We would then assume that the claims come from a heterogeneous group of clients, each one of them generating claims according to a Poisson distribution with the intensity varying from one group to another.

Modelling the risk process 26 Mixed Poisson process cont. If Ñ is a HPP with intensity 1 and Λ is a positive random variable independent of Ñ, then the process N = Ñ Λ = (Ñ(Λt)) t is called a mixed Poisson process. The random variable Λ is called a structure variable. A mixed Poisson process has stationary increments, however the independent increments condition is violated. The most common choice for the distribution of the structure variable Λ is the gamma distribution. In such a case the mixed Poisson proces is called a negative binomial process or Pólya process.

Modelling the risk process 27 Mixed Poisson process cont. In the mixed Poisson process the distribution of {N t } is given by a mixture of Poisson processes. Conditioning on the extrinsic random variable Λ, the process {N t } behaves like a HPP. Hence, the process can be generated in the following way: first a realization of a non-negative random variable Λ is generated, conditioned upon its realization, {N t } as a HPP with that realization as its intensity is constructed.

Modelling the risk process 28 Simulation of the mixed Poisson process Making the algorithm more formal we can write: Step 1: generate a realization λ of the random intensity Λ Step 2: set T 0 = 0 Step 3: for i = 1, 2,..., n do Step 3a: generate an exponential random variable E with intensity λ Step 3b: set T i = T i 1 + E

Modelling the risk process 29 Simulation of the mixed Poisson process cont. Since for each t the claim numbers {N t } up to time t are Poisson with intensity Λt, in the mixed case it is reasonable to consider the premium function of the form: c(t) = (1 + θ)µλt, where µ = EX k and θ > 0 is the relative safety loading. Then the risk process takes the form: R t = u + (1 + θ)µλt N t i=1 X i.

Modelling the risk process 30 Cox process The Cox process, or doubly stochastic Poisson process, provides flexibility by letting the intensity not only depend on time but also by allowing it to be a stochastic process. Cox processes seem to form a natural class for modeling risk and size fluctuations. IF Ñ is a HPP with intensity 1 and {Λ(t)} is a stochastic process with Λ = 0, non-decreasing sample paths and idependent of Ñ, then the process N = Ñ Λ = (Ñ(Λ)) is called a Cox process or doubly stochastic Poisson process.

Modelling the risk process 31 Cox process cont. The intensity process {Λ(t)} is used to generate another process {N t } by acting as its intensity. That is, {N t } is a Poisson process conditional on {Λ(t)} which itself is a stochastic process. If {Λ(t)} is deterministic, then {N t } is a NHPP. This property suggests a simulation method.

Modelling the risk process 32 Simulation of the Cox process Step 1: generate a realization λ(t) of the intensity process {Λ(t)} for a sufficiently large time period Step 2: set λ = max {λ(t)} Step 3: set T 0 = 0 and T = 0 Step 4: for i = 1, 2,..., n do Step 4a: generate an exponential random variable E with intensity λ Step 4b: set T = T + E Step 4c: generate a random variable U U(0, 1) Step 4d: if U > λ(t )/λ then return to step 4a ( reject the arrival time) else set T i = T ( accept the arrival time)

Modelling the risk process 33 Simulation of the Cox process cont. The premium function is a generalization of the former premium functions: c(t) = (1 + θ)µ t 0 Λ(s)ds, where µ = EX k and θ > 0 is the relative safety loading. Then the risk process takes the form: R t = u + (1 + θ)µ t 0 Λ(s)ds N t i=1 X i.

Modelling the risk process 34 Renewal process If the waiting times W i are i.i.d. and nonnegative then the resulting sequence is a renewal process. Note, that the HPP is a renewal process with exponentially distributed inter-arrival times. Hence, we can generate the arrival times of a renewal process by: Step 1: set T 0 = 0 Step 2: for i = 1, 2,..., n do Step 2a: generate a random variable X with an assumed distribution function F Step 2b: set T i = T i 1 + X

Modelling the risk process 35 Renewal process cont. For renewal claim arrival processes a constant premium rate allows for a constant safety loading. Let {N t } be a renewal process and assume that EW 1 = 1/λ <. Then the premium function is defined in a natural way as: c(t) = (1 + θ)µλt, like in the homogeneous Poisson process case.

Modelling the risk process 36 Empirical analysis Danish fire losses recorded by Copenhagen Re. Losses in profits connected with fires Loss sizes Lognormal, Pareto, Burr, gamma, Weibull and mixture of two exponentials distributions Claim counting process Homogeneous and nonhomogeneous process

Modelling the risk process 37 Losses (DKK million) 0 20 40 60 Log(1-F(x)) -2-1 0 1980 1985 1990 Time 0 4 8 12 16 20 Losses (DKK million) Figure 1: Left panel: Illustration of the major Danish fire losses adjusted for inflation. Right panel: Logarithm of the right tails of the empirical claim size distribution function (thick blue solid line) together with lognormal (red dotted line) and Burr (thin black solid line) fits.

Modelling the risk process 38 Danish fire losses. Loss sizes Losses (DKK million) 0 20 40 60 Log(1-F(x)) -2-1 0 1980 1985 1990 Time 0 4 8 12 16 20 Losses (DKK million) STFrisk03.xpl STFrisk04.xpl

Modelling the risk process 39 d.f.: Lognormal Pareto Burr Weibull Para- µ = 12.704 α = 2.4189 α = 0.8935 α = 0.6963 meters: σ = 1.4271 λ = 1.0261e6 λ = 1.1219e7 λ = 8.9740e-5 τ = 1.2976 χ 2 56.109 73.879 48.493 129.24 KS 0.0373 0.0397 0.0413 0.0783 CM 0.1687 0.2878 0.1438 1.5245 AD 1.0533 2.7712 0.8221 10.638

Modelling the risk process 40 Danish fire losses cont. Claim counting process Mean-value function 0 100 200 300 400 500 600 700 ACF 0 0.5 1 0 1 2 3 4 5 6 7 8 9 10 11 Time (years) 0 5 10 15 20 25 30 Time lag (qtr) STFrisk05.xpl STFrisk06.xpl

Modelling the risk process 41 Danish fire losses cont. Claim counting process The data reveals no seasonality A clear increasing trend can be observed in the number of quarterly losses We tested different exponential and polynomial functional forms A simple linear intensity function λ(s) = a + bs yielded the best fit Applying a least squares procedure we arrived at the values: a = 13.97 and b = 7.57.

Modelling the risk process 42 Danish fire losses cont. The fitted risk process We consider a hypothetical scenario where the insurance company insures losses resulting from fire damage The company s initial capital is assumed to be u = 100 million kr The relative safety loading used is θ = 0.5 We chose two models of the risk process: a non-homogeneous Poisson process with lognormal claim sizes and a non-homogeneous Poisson process with Burr claim sizes.

Modelling the risk process 43 Danish fire losses cont. The fitted risk process Capital (DKK million) 0 100 200 300 400 500 Capital (DKK million) 0 200 400 600 800 1000 1200 0 1 2 3 4 5 6 7 8 9 10 11 Time (years) 0 1 2 3 4 5 6 7 8 9 10 11 Time (years) STFrisk07.xpl STFrisk08.xpl