scholarly journals The Effect of Reinsurance on the Degree of Risk Associated with an Insurer's Portfolio

1980 ◽  
Vol 11 (2) ◽  
pp. 119-135 ◽  
Author(s):  
M. Andreadakis ◽  
H. R. Waters
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Poisson Distribution ◽  
Compound Poisson Distribution ◽  
Proportional Reinsurance ◽  
Insurance Portfolio ◽  
Degree Of Risk ◽  
Definition Of ◽  
Image Position ◽  
Annual Premium

There are many reasons why an insurer may choose to reinsure a part of his portfolio (see, for example, Carter (1979, p. 5 ff.)) and many ways in which he can assess the effectiveness of the reinsurance arrangements he makes. In this paper we assume the insurer wishes to reinsure a part of his portfolio in order to reduce its “riskiness”. We take as given a portfolio consisting of n independent risks together with the total premium charged to insure these risks and we investigate the effect on the degree of risk associated with the portfolio (see §3 for a definition) of varying the excess of loss or proportional reinsurance limits for each risk.We are given an insurance portfolio consisting of n independent risks. A risk may consist of a single policy or a group of policies: the essential points being that a reinsurance limit, either excess of loss or proportional, is the same for all claims arising from a particular risk, although reinsurance limits may vary from one risk to another. We assume the claims arising from each risk have a compound Poisson distribution. To be more precise, we assume the number of claims arising from the i-th risk is a Poisson process with mean Pi claims each year and the size of each claim has distribution function Fi. As usual, the size of a claim is independent of the time at which it occurs and of all other claims. We also assume that Fi(O) = O for each i, so that we consider only positive claims amounts. We take as given the total annual premium, P, charged by the insurer in respect of these risks. We make no assumption about the way in which P is calculated but we do assume that

scholarly journals Statistical Methodology for Large Claims

1977 ◽  
Vol 9 (1-2) ◽  
pp. 1-9 ◽  
Author(s):  
J. Tiago de Oliveira
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Critical Level ◽  
Compound Poisson Process ◽  
Distribution Functions ◽  
Asymptotic Distributions ◽  
Jump Function ◽  
Image Position ◽  
Asymptotic Forms ◽  
Independent And Identically Distributed

The question of large claims in insurance is, evidently, a very important one, chiefly if we consider it in relation with reinsurance. To a statistician it seems that it can be approached, essentially, in two different ways.The first one can be the study of overpassing of a large bound, considered to be a critical one. If N(t) is the Poisson process of events (claims) of intensity v, each claim having amounts Yi, independent and identically distributed with distribution function F(x), the compound Poisson processwhere a denotes the critical level, can describe the behaviour of some problems connected with the overpassing of the critical level. For instance, if h(Y, a) = H(Y − a), where H(x) denotes the Heavside jump function (H(x) = o if x < o, H(x) = 1 if x ≥ o), M(t) is then the number of claims overpassing a; if h(Y, a) = Y H(Y − a), M(t) denotes the total amount of claims exceeding the critical level; if h(Y, a) = (Y − a) H(Y − a), M(t) denotes the total claims reinsured for some reinsurance policy, etc.Taking the year as unit of time, the random variables M(1), M(2) − M(1), … are evidently independent and identically distributed; its distribution function is easy to obtain through the computation of the characteristic function of M(1). For details see Parzen (1964) and the papers on The ASTIN Bulletin on compound processes; for the use of distribution functions F(x), it seems that the ones developed recently by Pickands III (1975) can be useful, as they are, in some way, pre-asymptotic forms associated with tails, leading easily to the asymptotic distributions of extremes.

Convolution equivalence and infinite divisibility

2004 ◽  
Vol 41 (02) ◽  
pp. 407-424 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Distribution Function ◽  
Poisson Distribution ◽  
Infinite Divisibility ◽  
Compound Poisson Distribution ◽  
Infinitely Divisible Distributions ◽  
Random Sum ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Tail Behaviour ◽  
Equivalence Result

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

High density shot noise and gaussianity

1971 ◽  
Vol 8 (1) ◽  
pp. 118-127 ◽  
Author(s):  
A. Papoulis
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Shot Noise ◽  
Average Density ◽  
Normal Process ◽  
Noise Process ◽  
Shot Noise Process ◽  
Mean And Variance ◽  
Image Position ◽  
Random Times

The distance from Gaussianity of the shot noise process is considered, where ti are the random times of a Poisson process with average density λ(t). With F(x) the distribution function of x(t) and G(x) that of a normal process with the same mean and variance as x(t) it is shown that where If the process x(t) is stationary with λ(t) =λ and h(t, τ) = h(t – τ) and the function h(t) is bandlimited by ωc, then the above yields

scholarly journals Calculation of Ruin Probabilities when the Claim Distribution is Lognormal

1977 ◽  
Vol 9 (1-2) ◽  
pp. 231-246 ◽  
Author(s):  
Olof Thorin ◽  
Nils Wikstad
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Linear Combination ◽  
Lognormal Distribution ◽  
Ruin Probabilities ◽  
Exponential Distributions ◽  
Absolutely Continuous ◽  
Image Position

In this paper some ruin probabilities are calculated for an example of a lognormal claim distribution. For that purpose it is shown that the lognormal distribution function, Λ(y), may be written in the formwhere V(x) is absolutely continuous and without being a distribution function preserves some useful properties of such a function.An attempt is also made to give an approximant Λα(y) to Λ(y) such that Λα(y) is a linear combination of a low number of exponential distributions. For comparison, ruin probabilities are also calculated for two examples of Λα(y).In the considered numerical cases it is assumed that the occurrence of claims follows a Poisson process.

Convolution equivalence and infinite divisibility

2004 ◽  
Vol 41 (2) ◽  
pp. 407-424 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Distribution Function ◽  
Poisson Distribution ◽  
Infinite Divisibility ◽  
Compound Poisson Distribution ◽  
Infinitely Divisible Distributions ◽  
Random Sum ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Tail Behaviour ◽  
Equivalence Result

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

scholarly journals Comments on different deductions of expressions for conditional expectations

1969 ◽  
Vol 5 (2) ◽  
pp. 298-302
Author(s):  
Carl Philipson
Keyword(s):  
Distribution Function ◽  
Product Space ◽  
Arithmetic Mean ◽  
Least Square ◽  
Linear Estimate ◽  
Conditional Mean ◽  
Conditional Expectations ◽  
The Mean ◽  
Definition Of ◽  
Image Position

1. In a paper presented to the fifth Astin Colloquium (Lucerne, 1965) Bühlmann has given some propositions with regard to Experience Rating understood as a sequence of estimates of the expectation with respect to the distribution function of θ, H(θ) say, of μ(θ) which is the mean for fixed θ with respect to the distribution function G(x; θ) of the variables for each value of ν = 1, 2 … n. In such problems the estimator function for μ(θ) is generally chosen to be the conditional mean of μ(θ) for a given set of observed values of This is generally justified by the principle of least square deviation. According to Bühlmann this justification is not sufficient. Therefore, he bases the choice of this estimator function upon a postulate of equilibrium, described in the following lines.Let X′ be a subset of X, and C(X′) a cylinder with the base X′ ⊂ X in the product space X × Θ, where X is the set of all possible and Θ the set of all possible θ, then the postulate of equilibrium implies the equality between the expectations of μ(θ) and of a function of on each cylinder C(X′). This is exactly Kolmogoroff's definition of the conditional expectation: Bühlmann states, further, that the best linear estimate of (1) based on the arithmetic mean of n sample values xν of ν = 1, 2 … n can be written in the form of the Credibility Formula: where the symbol E[·] denotes the expectation over the product space X × Θ and If the are independent and identically distributed, this leads to where k = E [σ2 (θ)] / Var [ν(θ)], and σ2(θ) is the variance for a fixed θ of G(x; θ). This case of the credibility formula is, generally, applied in American practice. (The proposition has later been proved for more general conditions).

High density shot noise and gaussianity

1971 ◽  
Vol 8 (01) ◽  
pp. 118-127 ◽  
Author(s):  
A. Papoulis
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Shot Noise ◽  
Average Density ◽  
Normal Process ◽  
Noise Process ◽  
Shot Noise Process ◽  
Mean And Variance ◽  
Image Position ◽  
Random Times

The distance from Gaussianity of the shot noise processis considered, wheretiare the random times of a Poisson process with average densityλ(t).WithF(x) the distribution function ofx(t) andG(x) that of a normal process with the same mean and variance asx(t) it is shown thatwhereIf the processx(t) is stationary with λ(t) =λandh(t, τ) =h(t – τ) and the functionh(t) is bandlimited by ωc, then the above yields

A characterization of the Poisson process using forward recurrence times

1975 ◽  
Vol 78 (3) ◽  
pp. 513-516 ◽  
Author(s):  
Valerie Isham ◽  
D. N. Shanbhag ◽  
M. Westcott
Keyword(s):  
Distribution Function ◽  
Poisson Process ◽  
Real Line ◽  
Renewal Process ◽  
Recurrence Time ◽  
Recurrence Times ◽  
Arithmetic Distribution ◽  
Image Position ◽  
Forward Recurrence Time

Consider a renewal process on the nonnegative real line with non-arithmetic distribution function F(x). Denote by V(x; t) the distribution function of the forward recurrence time from t, t ≤ 0. If t is chosen at random with distribution function Ф(t), the corresponding unconditional forward recurrence time has distribution function

Log-Compound-Poisson Distribution Model of Intermittency in Turbulence

1998 ◽  
Vol 53 (10-11) ◽  
pp. 828-832
Author(s):  
Feng Quing-Zeng
Keyword(s):  
Energy Dissipation ◽  
Poisson Distribution ◽  
Turbulent Energy ◽  
Compound Poisson Distribution ◽  
Distribution Model ◽  
Velocity Difference ◽  
Scaling Exponents ◽  
Compound Poisson ◽  
Developed Turbulence ◽  
Experimental Values

Abstract The log-compound-Poisson distribution for the breakdown coefficients of turbulent energy dissipation is proposed, and the scaling exponents for the velocity difference moments in fully developed turbulence are obtained, which agree well with experimental values up to measurable orders. The under-lying physics of this model is directly related to the burst phenomenon in turbulence, and a detailed discussion is given in the last section.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

Sign in / Sign up

Export Citation Format

Share Document