Hans Bühlmann

Mathematical methods in risk theory

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Experience rating and credibility

Let me begin with some practical examples of experience rating. a) Swiss Automobile Tariff 1963 — Within each tariff-position there are 22 grades: — The new owner of a car starts at grade 9 — The basic premium is determined on the basis of objective characteristics of the risk and essentially depends on the horse-power classification of the car — The 22 grades are experience-rated as follows: For each accident one rises three grades and for each accident-free year one falls one grade. A driver who has I accident in every 4 years hence remains within four adjacent grades. b) Sliding Scale Premiums in Reinsurance Excess of Loss Contracts often stipulate that: The rate of premium to be applied to the subject premium volume is determined at the end of the cover period as follows: subject to a minimum of 0,04 and a maximum of 0,08 c) Participation in Mortality Profit in Group Life Insurance A group life insurance covers the members of the group on a one year term basis. It is often agreed that at the end of the year mortality profits are given back to the group according to the formula refund = x % gross premiums — y % claims (where x y ) All these examples fall under the heading “Experience Rating”. What do they have in common? Definition : A system by which the premium of the individual risk depends upon the claims experience of this same individual risk .

An Economic Premium Principle

(a) The notion of premium calculation principle has become fairly generally accepted in the risk theory literature. For completeness we repeat its definition:A premium calculation principle is a functional assigning to a random variable X (or its distribution function Fx(x)) a real number P. In symbolsThe interpretation is rather obvious. The random variable X stands for the possible claims of a risk whereas P is the premium charged for assuming this risk.This is of course formalizing the way actuaries think about premiums. In actuarial terms, the premium is a property of the risk (and nothing else), e.g.(b) Of course, in economics premiums are not only depending on the risk but also on market conditions. Let us assume for a moment that we can describe the risk by a random variable X (as under a)), describe the market conditions by a random variable Z.Then we want to show how an economic premium principlecan be constructed. During the development of the paper we will also give a clear meaning to the random variable Z:In the market we are considering agents i = 1, 2, …, n. They constitute buyers of insurance, insurance companies, reinsurance companies.Each agent i is characterized by hisutility function ui(x) [as usual: ]initial wealth wi.In this section, the risk aspect is modelled by a finite (for simplicity) probability space with states s = 1, 2, …, S and probabilities I€s of state s happening.

The General Economic Premium Principle

We give an extension of the Economic Premium Principle treated in Astin Bulletin, Volume 11 where only exponential utility functions were admitted. The case of arbitrary risk averse utility functions leads to similar quantitative results. The role of risk aversion in the treatment is essential. It also permits an easy proof for the existence of equilibrium.

Some Inequalities for Stop-Loss Premiums

In this paper any given risk S (a random variable) is assumed to have a (finite or infinite) mean. We enforce this by imposing E[S−] v((1−z)Q)} is not empty.Proof: a) b) Because of a) E[v(S−zQ)] is always finite or equal to + ∞ If v(− ∞) = − ∞ then E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q. The left hand side of the inequality is a nonincreasing continuous function in P (strictly decreasing if z > 0), while the right hand side is a nondecreasing continuous function in Q (strictly increasing if z > 1).If v(− ∞) = c finite then E[v(S − zQ)] > c(otherwise S would need to be equal to − ∞ with probability 1) and again E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q.

On Esscher Transforms in Discrete Finance Models

The object of our study iswhere each Sn is a m-dimensional stochastic (real valued) vector, i.e.denned on a probability space (I©, , P) and adapted to a filtration (n)0≤n≤N with 0 being the Iƒ-algebra consisting of all null sets and their complements. In this paper we interpret as the value of some financial asset k at time n.Remark: If the asset generates dividends or coupon payments, think of as to include these payments (cum dividend process). Think of dividends as being reinvested immediately at the ex-dividend price.Definition 1(a) A sequence of random vectorswhereis called a trading strategy. Since our time horizon ends at time N we must always have I‘N â‰i 0.The interpretation is obvious: stands for the number of shares of asset k you hold in the time interval [n,n + 1). You must choose I‘n at time n.(b) The sequence of random variableswhere Sn stands for the payment stream generated by I‘ (set I‘−1 â‰i 0).

Market-consistent actuarial valuation

Introduction.- Stochastic discounting.- Valuation portfolio in life insurance.- Financial risks.- Valuation portfolio in non-life insurance.- Selected topics.

Premium Calculation from Top Down

This paper is intended to show how premiums are related to the stability criterion imposed on a portfolio of risks and to the dividend requirements for the capital invested into the insurance operation. The point is that premium calculation should be seen as a consequence of the strategic concepts adopted by the insurance carrier.

Numerical evaluation of the compound Poisson distribution: Recursion or Fast Fourier Transform?

1. The problem The finite vector p=(p 1,p 2, ...,ps ) defines a probability distribution on the integers 1,2, ...,s.

THE MEAN SQUARE ERROR OF PREDICTION IN THE CHAIN LADDER RESERVING METHOD (MACK AND MURPHY REVISITED)

We revisit the famous Mack formula [2], which gives an estimate for the mean square error of prediction MSEP of the chain ladder claims reserving method: We define a time series model for the chain ladder method. In this time series framework we give an approach for the estimation of the conditional MSEP. It turns out that our approach leads to results that differ from the Mack formula. But we also see that our derivation leads to the same formulas for the MSEP estimate as the ones given in Murphy [4]. We discuss the differences and similarities of these derivations.