scholarly journals Minimum Variance Reinsurance

1962 ◽  
Vol 2 (2) ◽  
pp. 257-260 ◽  
Author(s):  
Stefan Vajda
Keyword(s):  
Minimum Variance ◽  
Point Of View ◽  
Stieltjes Integral ◽  
Optimum Amount ◽  
Natural Conditions ◽  
Excess Loss ◽  
Total Claim ◽  
Image Position ◽  
Stop Loss ◽  
Elegant Proof

In a paper entitled “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance” (XVIth Int. Congr. Act. Bruxelles 1960) Karl Borch has shown that, if the reinsurance premium is given, the smallest variance of the cedent's payments is obtained by a stop-loss reinsurance contract. Paul Markham Kahn, in “Some Remarks on a Recent Paper by Borch”, a paper read to the 1961 Astin Colloquium, has given an elegant proof of this theorem which appears to apply also to cases not considered by Borch. In this paper we study the problem from the reinsurer's point of view and it will be seen that, under natural conditions which are also used in the proof of the Borch-Kahn theorem, the minimum variance of the reinsurer's payments is obtained by a quota contract. This focusses attention on a peculiar opposition of interests of the two partners of a reinsurance contract. However, we do not enter any further into the investigation of a possible resolution of this conflict.We study a problem concerning the division of risk between a cedent and his reinsurer. The risk may refer to a whole portfolio (in which case one might consider a Stop-Loss contract), or to a single contract (when an Excess-Loss contract is a possibility). We shall here use the nomenclature of a portfolio reinsurance.Let it be assumed that a function F(x) is known which gives the probability of a total claim not exceeding x. We have then in Stieltjes integral notationThe two partners to a reinsurance arrangement agree that the reinsurer reimburses m(x).x out of a claim of x, where m(x) is a continuous and differentiate function of x and o ≤ m(x) ≤ 1.

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

The Representation of (C, k) Summable Series in Fourier Form

1978 ◽  
Vol 21 (2) ◽  
pp. 149-158 ◽  
Author(s):  
G. E. Cross
Keyword(s):  
Trigonometric Series ◽  
Point Of View ◽  
Perron Integral ◽  
Image Position

Several non-absolutely convergent integrals have been defined which generalize the Perron integral. The most significant of these integrals from the point of view of application to trigonometric series are the Pn- and pn-integrals of R. D. James [10] and [11]. The theorems relating the Pn -integral to trigonometric series state essentially that if the series1.1

scholarly journals Developing a Multi-channel Beamformer by Enhancing Spatially Constrained ICA for Recovery of Correlated EEG Sources

2018 ◽  
Author(s):  
N Samadzadehaghdam ◽  
B MakkiAbadi ◽  
E Eqlimi ◽  
F Mohagheghian ◽  
H Khajehpoor ◽  
...  
Keyword(s):  
Mean Squared Error ◽  
Partial Information ◽  
Minimum Variance ◽  
Point Of View ◽  
The Body ◽  
Correlated Sources ◽  
Source Imaging ◽  
Neuron Populations ◽  
Eeg Data ◽  
Robust Beamformer

Background: Brain source imaging based on electroencephalogram (EEG) data aims to recover the neuron populations’ activity producing the scalp potentials. This procedure is known as the EEG inverse problem. Recently, beamformers have gained a lot of consideration in the EEG inverse problem.Objective: Beamformers lack acceptable performance in the case of correlated brain sources. These sources happen when some regions of the brain have simultaneous or correlated activities such as auditory stimulation or moving left and right extremities of the body at the same time. In this paper, we have developed a multichannel beamformer robust to correlated sources. Material and Methods: We have looked at the problem of brain source imaging and beamforming from a blind source separation point of view. We focused on the spatially constraint independent component analysis (scICA) algorithm, which generally benefits from the pre-known partial information of mixing matrix, and modified the steps of the algorithm in a way that makes it more robust to correlated sources. We called the modified scICA algorithm Multichannel ICA based EEG Beamformer (MIEB).Results: We evaluated the proposed algorithm on simulated EEG data and compared its performance quantitatively with three algorithms: scICA, linearly-constrained minimum-variance (LCMV) and Dual-Core beamformers; it is considered that the latter is specially designed to reconstruct correlated sources.Conclusion:The MIEB algorithm has much better performance in terms of normalized mean squared error in recovering the correlated/uncorrelated sources both in noise free and noisy synthetic EEG signals. Therefore, it could be used as a robust beamformer in recovering correlated brain sources. 

scholarly journals Some aspects on reinsurance profits and loadings

1971 ◽  
Vol 5 (3) ◽  
pp. 314-327 ◽  
Author(s):  
G. Benktander
Keyword(s):  
Operations Research ◽  
Coming Out ◽  
Simple Formula ◽  
Affirmative Answer ◽  
Point Of View ◽  
Excess Loss ◽  
Long Run ◽  
Loading System ◽  
Loading Problem ◽  
In The Beginning

In a note on the security loading of excess loss rates I am deducing a simple formula intended to replace some tedious calculations. In the beginning of that note I made the point that some authors recommend a loading proportional to the dispersion of the total claims amount of a treaty δ1 while others tend to favour .I also stated that a loading proportional to δ1 or its estimate δ1* could be deduced from the statistical uncertainty in measuring the risk (section 4).The question has been raised if and to what extent a loading system based on the dispersion is unduly punishing the smaller portfolios. This will be examined below.The pricing concept will be analyzed from the point of view of a big dominating Reinsurer who wants to be fair in all directions. The conclusion of this study supports an affirmative answer to the question put above.In a second part the loading is studied from a different angle bringing competition into the picture. The pricing or loading becomes a problem of operations research under the simplified assumption that profit is the only purpose of our activity. Not unexpectedly, the loading coming out from this aspect differs from those of part one.Part two also deals with the question of how much of the loadings which we are aiming for, get lost in the competitive process. It is also shown that in most cases the harder the competition is, the higher loadings shall be used.Part one and part two thus deal with the loading problem from different aspects, and illustrate the complexity of the problem. It is my hope that this note could stimulate further researches in this interesting and important area, also in a moment when some reinsurers are more concerned with the question of surviving than in fixing the loadings which should on the average and in the long run turn up as profits.

scholarly journals The Recent Nature of the Siberian Pole of Cold

1961 ◽  
Vol 3 (30) ◽  
pp. 1089-1096
Author(s):  
I. P. Gerasimov
Keyword(s):  
Late Glacial ◽  
Climatic Conditions ◽  
Point Of View ◽  
Natural Conditions ◽  
Dominant Type ◽  
North East ◽  
Natural Landscapes ◽  
Mountain System ◽  
Soil Temperatures ◽  
Extreme North

Abstract The Siberian pole of cold is situated in the extreme north-east of Eurasia (in the region of the Cherskiy mountain system, in the upper parts of the basins of the Yana, Indigirka and Kolyma Rivers). Particularly low air and soil temperatures have been observed in the intermontane areas. Among these localities is the famous Oymyakon, where the lowest minimum temperature in the Northern Hemisphere has been recorded. In the climate of this area extreme aridity, connected with the intracontinental position of the territory, is combined with intense cold. In the two highest massifs (Ulakhan-Chistay and Suntar-Khayata) small centres of recent glacierization (chiefly kars) are developed; there are also distinct traces of a more extensive older mountain glaciation. In the intermontane areas and on the principal level of the dissected hilly peneplain positive indications of a former glaciation are absent. However, the recent cryogenic phenomena represented by fossil ice, permafrost, taryns, as well as thermokarstic, solifluction and congelation features, are very abundant and diverse. The widespread development of all these features gives this territory a periglacial aspect, and also provides the possibility of using the study of many recent phenomena for palaeogeographical purposes. From this point of view, the processes leading to the formation of loess deposits (cryogenic facies) and the formation of structural and thixotropic soils arc of particular interest. The recent natural landscapes in this region are represented by a dominant type of larch tundra–forest associated with comparatively typical taiga bog formations in the depressions and xero-cryophile meadow–steppe landscapes on the steeper and warmer southern slopes. Such a unique landscape combination connected with the specific climatic conditions of this region provide a basis for interpreting the recent natural conditions of the Siberian pole of cold as a survival of the “late glacial.” At present these natural conditions are being intensively developed economically.

The Problem of Relativity in reference to several bodies

1926 ◽  
Vol 23 (3) ◽  
pp. 191-197
Author(s):  
R. Hargreaves
Keyword(s):  
Relative Motion ◽  
Point Of View ◽  
Image Position ◽  
Primary Form

§ 1. If the kinetic potential for the relative motion of two masses is written with an added constant asa close connexion with the relativity quadratic appears. The latter is in factwhere a modification of the primary formwhich shows an unaltered determinant. The condition in respect to the determinant, suggested, I believe, by Schwarzschild, is one which to me appears to give the most significant form to the results. From the dynamical standpoint we may regard it as imposing a counterpoise in the inertia coefficients to the modification introduced by the potential; or from a geometrical point of view we may regard it as minimizing the departure from the normal use of coordinates. An illuminating example of the loss of meaning that accompanies transformation in which this condition is disregarded is furnished by the isotropic form which is sometimes given to Einstein's quadratic.

scholarly journals The Recent Nature of the Siberian Pole of Cold

1961 ◽  
Vol 3 (30) ◽  
pp. 1089-1096
Author(s):  
I. P. Gerasimov
Keyword(s):  
Late Glacial ◽  
Climatic Conditions ◽  
Point Of View ◽  
Natural Conditions ◽  
Dominant Type ◽  
North East ◽  
Natural Landscapes ◽  
Mountain System ◽  
Soil Temperatures ◽  
Extreme North

AbstractThe Siberian pole of cold is situated in the extreme north-east of Eurasia (in the region of the Cherskiy mountain system, in the upper parts of the basins of the Yana, Indigirka and Kolyma Rivers). Particularly low air and soil temperatures have been observed in the intermontane areas. Among these localities is the famous Oymyakon, where the lowest minimum temperature in the Northern Hemisphere has been recorded. In the climate of this area extreme aridity, connected with the intracontinental position of the territory, is combined with intense cold.In the two highest massifs (Ulakhan-Chistay and Suntar-Khayata) small centres of recent glacierization (chiefly kars) are developed; there are also distinct traces of a more extensive older mountain glaciation. In the intermontane areas and on the principal level of the dissected hilly peneplain positive indications of a former glaciation are absent. However, the recent cryogenic phenomena represented by fossil ice, permafrost, taryns, as well as thermokarstic, solifluction and congelation features, are very abundant and diverse.The widespread development of all these features gives this territory a periglacial aspect, and also provides the possibility of using the study of many recent phenomena for palaeogeographical purposes. From this point of view, the processes leading to the formation of loess deposits (cryogenic facies) and the formation of structural and thixotropic soils arc of particular interest.The recent natural landscapes in this region are represented by a dominant type of larch tundra–forest associated with comparatively typical taiga bog formations in the depressions and xero-cryophile meadow–steppe landscapes on the steeper and warmer southern slopes. Such a unique landscape combination connected with the specific climatic conditions of this region provide a basis for interpreting the recent natural conditions of the Siberian pole of cold as a survival of the “late glacial.” At present these natural conditions are being intensively developed economically.

scholarly journals Approximations of the generalised Poisson function

1969 ◽  
Vol 5 (2) ◽  
pp. 213-226 ◽  
Author(s):  
Lauri Kauppi ◽  
Pertti Ojantakanen
Keyword(s):  
Distribution Function ◽  
Fixed Time ◽  
Risk Theory ◽  
Normal Distribution Function ◽  
Expected Number ◽  
Time Period ◽  
Poisson Function ◽  
Total Claim ◽  
Basic Functions ◽  
Image Position

One of the basic functions of risk theory is the so-called generalised Poisson function F(x), which gives the probability that the total amount of claims ξ does not exceed some given limit x during a year (or during some other fixed time period). For F(x) is obtained the well known expansion where n is the expected number of claims during this time period and Sk*(x) is the k:th convolution of the distribution function S(z) of the size of one claim. The formula (1) is, however, much too inconvenient for numerical computations and for most other applications. One of the main problems of risk theory, which is still partly open, is to find suitable methods to compute, or at least to approximate, the generalised Poisson function.A frequently used approximation is to replace F(x) by the normal distribution function having the same mean and standard deviation as F as follows: where α1 and α2 are the first zero-moments of S(z): SM(Z) is here again the distribution function of the size of one claim. To obtain more general results a reinsurance arrangement is assumed under which the maximum net retention is M. Hence the portfolio on the company's own retention is considered. If the reinsurance is of Excess of Loss type, then where S(z) is the distribution function of the size of one total claim.

scholarly journals Some Problems Connected with the Calculation of Stop Loss Premiums for Large Portfolios

1967 ◽  
Vol 4 (2) ◽  
pp. 170-174 ◽  
Author(s):  
Fredrik Esscher
Keyword(s):  
Distribution Function ◽  
Expected Number ◽  
Limit Values ◽  
Empirical Determination ◽  
The Mean ◽  
Premium Rates ◽  
Image Position ◽  
Stop Loss ◽  
The Given

When experience is insufficient to permit a direct empirical determination of the premium rates of a Stop Loss Cover, we have to fall back upon mathematical models from the theory of probability—especially the collective theory of risk—and upon such assumptions as may be considered reasonable.The paper deals with some problems connected with such calculations of Stop Loss premiums for a portfolio consisting of non-life insurances. The portfolio was so large that the values of the premium rates and other quantities required could be approximated by their limit values, obtained according to theory when the expected number of claims tends to infinity.The calculations were based on the following assumptions.Let F(x, t) denote the probability that the total amount of claims paid during a given period of time is ≤ x when the expected number of claims during the same period increases from o to t. The net premium II (x, t) for a Stop Loss reinsurance covering the amount by which the total amount of claims paid during this period may exceed x, is defined by the formula and the variance of the amount (z—x) to be paid on account of the Stop Loss Cover, by the formula As to the distribution function F(x, t) it is assumed that wherePn(t) is the probability that n claims have occurred during the given period, when the expected number of claims increases from o to t,V(x) is the distribution function of the claims, giving the conditioned probability that the amount of a claim is ≤ x when it is known that a claim has occurred, andVn*(x) is the nth convolution of the function V(x) with itself.V(x) is supposed to be normalized so that the mean = I.

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