Colin M. Ramsay

The Distribution of Sums of Certain I.I.D. Pareto Variates

ABSTRACT Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.

A solution to the ruin problem for Pareto distributions

Abstract An expression is derived for Ψ ( u ), the probability of ultimate ruin given an initial reserve of u in the case of a Pareto distribution. Laplace transforms and exponential integrals are used to derive this expression, which involves a single integral of real valued functions along the positive real line. Most importantly, the integrand is not of an oscillating kind. This expression for Ψ ( u ) is new, and may be used to form the basis of a more refined set of asymptotic approximations to Ψ ( u ). Finally, it shown that Ψ ( u ) can be expressed as the expected value of a function of a two parameter gamma random variable.

A Note on Random Survivorship Group Benefits

Consider a group of n independent lives age x where each life puts § 1 in a fund at time 0. The fund earns interest at rate i , and at the end of t years the accumulated value of the fund is divided equally among the survivors. The traditional approach to calculating the expected lump sum benefit per survivor from the initial group of n lives is based on the concept of a deterministic survivorship group. This approach ignores the stochastic nature of the survivorship process. In reality, the benefit per survivor is actually a random variable with an expected value which depends on the first inverse moment of a positive binomial random variable. Using Grabs and Savages (1954) recursive formula for the first inverse moment, it is shown that the traditional approach yields a fairly accurate approximation to the solution even when one assumes a random number of survivors.

Exact waiting time and queue size distributions for equilibrium M/G/1 queues with Pareto service

Abstract This paper solves the problem of finding exact formulas for the waiting time cdf and queue length distribution of first-in-first-out M/G/1 queues in equilibrium with Pareto service. The formulas derived are new and are obtained by directly inverting the relevant Pollaczek-Khinchin formula and involve single integrals of non-oscillating real valued functions along the positive real line. Tables of waiting time and queue length probabilities are provided for certain parameter values under heavy traffic conditions.

Calculating Ruin Probabilities via Product Integration

When claims in the compound Poisson risk model are from a heavy-tailed distribution (such as the Pareto or the lognormal), traditional techniques used to compute the probability of ultimate ruin converge slowly to desired probabilities. Thus, faster and more accurate roethods are needed. Product integration can be used in such situations to yield fast and accurate estimates of ruin probabilities because it uses quadrature weights that are suited to the underlying distribution. Tables of ruin probabilities for the Pareto and lognormal distributions are provided.

The distribution of compound sums of Pareto distributed losses

An expression is derived for the cumulative distribution function of , the aggregate losses in a period, where N is the random number of losses and the X k s are independent and identically distributed. Pareto random variables. Specific expressions are derived for the two most commonly used compound models in actuarial risk theory: the compound Poisson and the compound negative binomial.

The Distribution of Sums of I.I.D. Pareto Random Variables with Arbitrary Shape Parameter

Laplace transforms are used to derive an exact expression for the cdf of the sum of n i.i.d. Pareto random variables with common pdf f(x) = (α/β)(1 + x/β)−α−1 for x > 0, where α > 0 and is not an integer, and β > 0. An attractive feature of this expression is that it involves an integral of non oscillating real-valued functions on the positive real line. Examples of values of cdfs are provided and are compared to those determined via simulations.

On an integral equation for discounted compound-annuity distributions.

We consider a risk generating claims for a period of N consecutive years (after which it expires), N being an integer valued random variable. Let X k denote the total claims generated in the k th year, k >_ 1. The Xks are assumed to be independent and identically distributed random variables, and are paid at the end of the year. The aggregate discounted claims generated by the risk until it expires is defined as SN(V) = Z~=l vkXk, where v is the discount factor. An integral equation similar to that given by PANJER (1981) is developed for the pdfof SN(V). This is accomplished by assuming that N belongs to a new class of discrete distributions called annuity distributions. The probabilities in annuity distributions satisfy the following recursion: P n = P n I a + , for n = 1,2 . . . . . where an is the present value of an n-year immediate annuity.

A formula for the after‐tax APR for home mortgages

Purpose – The purpose of this paper is to provide a simple formula for determining a borrowers APR for mortgage loans after including the effects of mortgage interest tax deductions.Design/methodology/approach – This formula is derived by adjusting the APR provided by the lender for the length of the mortgage, the amount of discount points, the mortgage interest rate, and the borrowers tax rate.Findings – From this formula, it is found that the tax‐adjusted APR is not only lower but also more informative than the traditional APR.Research limitations/implications – The tax‐deductible items of a mortgage loan used in this formula for after‐tax APR are based on those stipulated under US tax law. However, this formula can easily be adapted to other internal rate of return methods such as the annual effective rate of return (AER). In addition, further research is needed to develop a formula when a mortgage loan is the result of a refinance. Under this situation, mortgage discount points are no longer tax‐ded...

Annuity distributions: A new class of compound Poisson distributions

Abstract A discrete random variable N is said to have an annuity distribution if its probabilities satisfy the recursion p n = p n -1 ( a + b/c n ), n = 1,2,3,… where a and b are real constants and c n = (1-e − n δ )/(e δ - 1), −∞ N satisfies a functional equation. This functional equation is used to prove that when N is an unbounded random variable then it is a compound Poisson random variable. It is also shown that the cumulants of N can be expressed as differences of Lambert series.