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.

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 On Optimal Properties of the Stop Loss Reinsurance

1967 ◽  
Vol 4 (2) ◽  
pp. 175-176 ◽  
Author(s):  
Erkki Pesonen
Keyword(s):  
Distribution Function ◽  
Consistency Condition ◽  
Random Variable ◽  
Insurance Company ◽  
Optimal Reinsurance ◽  
Conditional Expectations ◽  
Image Position ◽  
Stop Loss

Let F(x) be the distribution function of the total amount ξ of claims of an insurance company. It is assumed that this company reduces its liability by means of one or several reinsurance arrangements. Let the remaining part of claims amount be equal to η, which is another random variable. Reinsurance arrangements are supposed to fulfil the following consistency condition: if ξ = x, then almost certainly o ≤ η ≤ x. Presumably all reinsurance arrangements occuring in practice can be supposed to fulfil this requirement.The problem is to find an optimal reinsurance arrangement, i.e. an optimal random variable η, in the sense that, the net reinsurance premium being given, the variance of η reaches its minimum. In other words, a variable η is looked for, which givesE{η} = P = const.; V{η} = minimumIn the sequel we use conditional expectations in the sense defined by DOOB ([I]).Let η be an arbitrary random variable satisfying the consistency condition. IfR(x) = E{η|ξ =x},then evidently ≤ R(x) ≤ x. Further we have For the variance the inequality holds true, since Clearly the arrangement E {η|ξ} gives also to the reinsurer a smaller variance than the original arrangement.

scholarly journals Finite Sum Evaluation of the Negative Binomial-Exponential Model

1981 ◽  
Vol 12 (2) ◽  
pp. 133-137 ◽  
Author(s):  
Harry H. Panjer ◽  
Gordon E. Willmot
Keyword(s):  
Distribution Function ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Random Variable ◽  
Exponential Model ◽  
Compound Binomial ◽  
Binomial Parameter ◽  
Image Position ◽  
Finite Sum ◽  
Stop Loss

The compound negative binomial distribution with exponential claim amounts (severity) distribution is shown to be equivalent to a compound binomial distribution with exponential claim amounts (severity) with a different parameter. As a result of this, the distribution function and net stop-loss premiums for the Negative Binomial-Exponential model can be calculated exactly as finite sums if the negative binomial parameter α is a positive integer.The result is a generalization of Lundberg (1940).Consider the distribution ofwhere X1, X2, X3, … are independently and identically distributed random variables with common exponential distribution functionand N is an integer valued random variable with probability functionThen the distribution function of S is given byIf MX(t), MN(t) and MS(t) are the associated moment generating functions, then

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.

scholarly journals A generalized unimodality

1970 ◽  
Vol 7 (1) ◽  
pp. 21-34 ◽  
Author(s):  
Richard A. Olshen ◽  
Leonard J. Savage
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Stieltjes Transform ◽  
Real Random Variable ◽  
Unimodal Distribution ◽  
Image Position

Khintchine (1938) showed that a real random variable Z has a unimodal distribution with mode at 0 iff Z ~ U X (that is, Z is distributed like U X), where U is uniform on [0, 1] and U and X are independent. Isii ((1958), page 173) defines a modified Stieltjes transform of a distribution function F for w complex thus: Apparently unaware of Khintchine's work, he proved (pages 179-180) that F is unimodal with mode at 0 iff there is a distribution function Φ for which . The equivalence of Khintchine's and Isii's results is made vivid by a proof (due to L. A. Shepp) in the next section.

scholarly journals Discussion of Christofides' Conjecture Regarding Wang's Premium Principle

1999 ◽  
Vol 29 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Virginia R. Young
Keyword(s):  
Distribution Function ◽  
Proportional Hazards ◽  
Random Variable ◽  
Reliable Measure ◽  
Scale Family ◽  
Special Cases ◽  
Parametric Families ◽  
Image Position ◽  
Special Case ◽  
Family Of Distributions

Christofides (1998) studies the proportional hazards (PH) transform of Wang (1995) and shows that for some parametric families, the PH premium principle reduces to the standard deviation (SD) premium principle. Christofides conjectures that for a parametric family of distributions with constant skewness, the PH premium principle reduces to the SD principle. I will show that this conjecture is false in general but that it is true for location-scale families and for certain other families.Wang's premium principle has been established as a sound measure of risk in Wang (1995, 1996), Wang, Young, and Panjer (1997), and Wang and Young (1998). Determining when the SD premium principle is a special case of Wang's premium principle is important because it will help identify circumstances under which the more easily applied SD premium principle is a reliable measure of risk.First, recall that a distortion g is a non-decreasing function from [0, 1] onto itself. Wang's premium principle, with a fixed distortion g, associates the following certainty equivalent with a random variable X, (Wang, 1996) and (Denneberg, 1994):in which Sx is the decumulative distribution function (ddf) of X, Sx(t) = Pr(X > t), t ∈ R. If g is a power distortion, g(p) = pc, then Hg is the proportional hazards (PH) premium principle (Wang, 1995).Second, recall that a location-scale family of ddfs is , in which Sz is a fixed ddf. Alternatively, if Z has ddf Sz, then {X = μ + σZ: μ∈ R, σ > 0} forms a location-scale family of random variables, and the ddf of . Examples of location-scale families include the normal, Cauchy, logistic, and uniform families (Lehmann, 1991, pp. 20f). In the next proposition, I show that Wang's premium principle reduces to the SD premium principle on a location-scale family. Christofides (1998) observes this phenomenon in several special cases.

Smallest-fit selection of random sizes under a sum constraint: weak convergence and moment comparisons

1999 ◽  
Vol 31 (1) ◽  
pp. 178-198 ◽  
Author(s):  
Frans A. Boshuizen ◽  
Robert P. Kertz
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Continuous Mapping ◽  
Random Variables ◽  
Random Measure ◽  
Renewal Theory ◽  
Random Variable ◽  
Brownian Bridges ◽  
On Line ◽  
Image Position

In this paper, in work strongly related with that of Coffman et al. [5], Bruss and Robertson [2], and Rhee and Talagrand [15], we focus our interest on an asymptotic distributional comparison between numbers of ‘smallest’ i.i.d. random variables selected by either on-line or off-line policies. Let X1,X2,… be a sequence of i.i.d. random variables with distribution function F(x), and let X1,n,…,Xn,n be the sequence of order statistics of X1,…,Xn. For a sequence (cn)n≥1 of positive constants, the smallest fit off-line counting random variable is defined by Ne(cn) := max {j ≤ n : X1,n + … + Xj,n ≤ cn}. The asymptotic joint distributional comparison is given between the off-line count Ne(cn) and on-line counts Nnτ for ‘good’ sequential (on-line) policies τ satisfying the sum constraint ∑j≥1XτjI(τj≤n) ≤ cn. Specifically, for such policies τ, under appropriate conditions on the distribution function F(x) and the constants (cn)n≥1, we find sequences of positive constants (Bn)n≥1, (Δn)n≥1 and (Δ'n)n≥1 such that for some non-degenerate random variables W and W'. The major tools used in the paper are convergence of point processes to Poisson random measure and continuous mapping theorems, strong approximation results of the normalized empirical process by Brownian bridges, and some renewal theory.

Characterizations of certain multivariate distributions

1974 ◽  
Vol 75 (2) ◽  
pp. 219-234 ◽  
Author(s):  
Y. H. Wang
Keyword(s):  
Distribution Function ◽  
Normal Distribution ◽  
Random Variable ◽  
Sample Variance ◽  
Sufficient Condition ◽  
Multivariate Distributions ◽  
Sample Mean ◽  
One Dimensional ◽  
Independent Observations ◽  
Image Position

Let X1, X2, …, Xn, be n (n ≥ 2) independent observations on a one-dimensional random variable X with distribution function F. Letbe the sample mean andbe the sample variance. In 1925, Fisher (2) showed that if the distribution function F is normal then and S2 are stochastically independent. This property was used to derive the student's t-distribution which has played a very important role in statistics. In 1936, Geary(3) proved that the independence of and S2 is a sufficient condition for F to be a normal distribution under the assumption that F has moments of all order. Later, Lukacs (14) proved this result assuming only the existence of the second moment of F. The assumption of the existence of moments of F was subsequently dropped in the proofs given by Kawata and Sakamoto (7) and by Zinger (27). Thus the independence of and S2 is a characterizing property of the normal distribution.

On a three-state sojourn time problem

1971 ◽  
Vol 8 (04) ◽  
pp. 716-723 ◽  
Author(s):  
A. E. Gibson ◽  
B. W. Conolly
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Sojourn Time ◽  
Random Variable ◽  
The Real ◽  
Time Problem ◽  
Arbitrary Space ◽  
Image Position

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t 0 at which the visit to state T begins.

scholarly journals The Esscher Premium Principle: A Criticism

1981 ◽  
Vol 12 (1) ◽  
pp. 77-78 ◽  
Author(s):  
Benjamin Zehnwirth
Keyword(s):  
Loss Function ◽  
Utility Theory ◽  
Random Quantity ◽  
Present Note ◽  
Random Variable ◽  
Point Of View ◽  
The Other ◽  
Important Distinction ◽  
Image Position ◽  
Insight Into

The Esscher premium principle has recently had some exposure, namely, with the works of Bühlmann (1980) and Gerber (1980).Bühlmann (1980) devised the principle and coined the name for it within the framework of utility theory and risk exchange. Geruber (1980), on the other hand, gives further insight into the principle by studying it within the realm of forecasting in much the same spirit as credibility theorists forecast premiums. However, there is an important distinction: the choice of loss function.The present note sets out to criticize this relatively embryonic principle using decision theoretic arguments and indicates that the Esscher premium is essentially a small perturbation of the well established linearized credibility premium Bühlmann (1970).Let H denote the Esscher premium principle with loading h > o. That is, if X is an observable random variable and Y is a parameter (a risk or a random quantity) to be forecasted then the Esscher premium is given byThat is, H(Y ∣ X) is the Bayes decision rule for estimating Y given the data X relative to the loss functionwhere a is the estimate of Y, and of course the loading h is greater than zero.Now, for the clincher. This loss function is nonsensical from the point of view of estimation. It indicates a loss (or error) to the forecaster that is essentially the antithesis of relative loss.

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