The model and assumptions

192 C.V. Christensen, H. Schmidli Insurance: Mathematics and Economics 27 2000 189–200 occurred till time T and Π T are the premiums earned to cover the claims occurring till time T . In our model Π T would be a linear function. This would imply that the process L ∞ T − Π T : T ≥ 0 is a martingale under the pricing measure. This condition will determine the risk aversion coefficient α; see for instance, Sondermann 1988. To proceed further in the calculation of the future price, one has to choose a model for L t . Aase 1994 used a compound Poisson model. This can be seen as catastrophes occurring at certain times and claims are reported immediately. In such a model there would not be a need for the prolonged reporting period. In Embrechts and Meister 1997, a doubly stochastic Poisson model is introduced. Here, a high intensity level will occur shortly after a catastrophe, where more claims are expected to be reported. In Klüppelberg and Mikosch 1997, Example 5.3, the asymptotic expected value and asymptotic variance for a general compound process are obtained. The aim of this paper is to model the claims reported to the companies as individual claims with a reporting lag. This is done by modelling the aggregate claim from a single catastrophe as a compound mixed Poisson model. We thereby obtain the possibility to separate the individual claims and to model the reporting times of the claims. In Section 3.1, we calculate the future price using a compound Poisson model, whereas in Section 3.2, the results are extended by using a compound negative binomial model, represented as a mixed compound Poisson model. We thereby can estimate the mixing parameter from the reporting flow.

2. The model and assumptions

Let T 1 denote the end of the event period and T 2 T 1 the end of the reporting period. We work on a complete probability space Ω, F , P containing the following random variables and stochastic processes: L t the aggregate amount of reported claims till time t N t the number of catastrophes occurred in the interval [0, t] M i the number of claims from the ith catastrophe M i t the number of claims from catastrophe i reported until t Y ij the claim size for the j th claim from the ith catastrophe D ij the reporting lag for the j th claim from the ith catastrophe τ i the occurrence time of the ith catastrophe We assume the following: • F t is the smallest right continuous complete filtration, such that the aggregate amount of reported losses L t at time t is F t-adapted. • N t is a Poisson process with rate Λ ∈ 0, ∞. • M i : i ∈ N, N t : 0 ≤ t ≤ T 1 , D ij : i, j ∈ N, Y ij : i, j ∈ N are independent. • M i is the mixed Poisson distributed with mixing distribution F λ . That is, there are random variables λ i with distribution F λ such that, given λ i , M i is conditionally Poisson distributed with parameter λ i . If the distribution F λ is degenerated λ i = λ for some constant λ the unconditional distribution of M i is Poisson with para- meter λ. • λ i : i ∈ N are i.i.d. and independent of N t , D ij , Y ij . • D ij ∼ F D , Y ij ∼ F Y . We denote by Y D, respectively a generic variable for Y ij D ij , and by m Y r = E [e rY ] the moment generating function of the claim sizes. • the j th claim Y ij from the ith catastrophe is reported at time τ i + D ij . We have N T −N t ∼ PoiΛT −t and τ N t + 1 , . . . , τ N T |N T −N t = n has the same distribution as U 1 , . . . , U n where the U i are i.i.d. uniformly distributed on the interval [t, T ] and U i denotes the order statistics; see for instance, Rolski et al. 1999, Theorem 5.2.1. Moreover, it can be shown, which may seem a little bit surprising, that the number of claims M i T 2 − M i t from catastrophe i reported in the period [t, T 2 ], given λ i , is conditionally independent of the number of claims M i t reported in the period [τ i , t ]. C.V. Christensen, H. Schmidli Insurance: Mathematics and Economics 27 2000 189–200 193 Moreover, for 1 ≤ i ≤ N T 1 , given τ i , λ i , we have i ≤ N t : M i t | λ i ,τ i ∼ Poiλ i F D t − τ i , 2.1 and i N t : M i T 2 − M i t | λ i ,τ i ∼ Poiλ i F D T 2 − τ i − F D t − τ i , M i T 2 − M i t | λ i ,τ i ∼ Poiλ i F D T 2 − τ i . In our model the claims Y ij from the ith catastrophe are randomly ordered. This simplifies the modelling of the reporting lags D ij . Let D i :j : 1 ≤ j ≤ M i be the order statistics of the D ij 1≤j ≤M i , and Y i :j be the claim corresponding to D i :j . Then the claims occurred before T 1 and reported till t ≤ T 2 amount to L t = N t ∧T1 X i= 1 M i t X j = 1 Y i :j . In particular, the final aggregate amount L T 2 can be represented as L T 2 = L t + N t ∧T1 X i= 1 M i T 2 X j =M i t + 1 Y i :j + N T1 X i=N t ∧T1 + 1 M i T 2 X j = 1 Y i :j . For the rest of this section we work with the measure P conditioned on F t . Let S i = M i T 2 X j =M i t + 1 Y i :j . For i ≤ N t , given λ i , S i is then compound Poisson distributed with intensity parameter λ i F D T 2 − τ i − F D t − τ i . At time t, N t is known, so S 1 + · · · + S N t conditioned on λ 1 , . . . , λ N t is again compound Poisson distributed with parameter λ 1 F D T 2 − τ 1 − F D t − τ 1 + · · · + λ N t F D T 2 − τ N t − F D t − τ N t . The latter is known from risk theory; see for instance Gerber 1979, p. 13 or Rolski et al. 1999, Theorem 4.2.2. For N t i ≤ N T 1 , given λ i and τ i , S i is then compound Poisson distributed with intensity parameter λ i F D T 2 − τ i . We again have that S N t + 1 + · · · + S N T1 conditioned on N T 1 , λ N t + 1 , . . . , λ N T1 and τ N t + 1 , . . . , τ N T1 , is compound Poisson distributed with intensity parameter P N T1 i=N t + 1 λ i F D T 2 − τ i . So all in all we get that L T 2 − L t = S 1 + · · · + S N t + S N t + 1 + · · · + S N T1 given N T 1 , λ 1 , . . . , λ N T1 , τ 1 , . . . , τ N T1 is compound Poisson distributed with intensity parameter N t X i= 1 λ i F D T 2 − τ i − F D t − τ i + N T1 X i=N t + 1 λ i F D T 2 − τ i d = N t X i= 1 λ i F D T 2 − τ i − F D t − τ i + N T1 X i=N t + 1 λ i F D T 2 − ˜ τ i 2.2 where ˜ τ i are i.i.d. uniformly distributed on t, T 1 and independent of F t . Here d = means equality in distribution. Thus, for t fixed L T 2 − L t becomes a mixed compound Poisson model. 194 C.V. Christensen, H. Schmidli Insurance: Mathematics and Economics 27 2000 189–200

3. Calculation of the CAT-future price