Fourier inversion—distribution functions—long tails

Abstract

Given a characteristic functionφ(t) we want to calculate the corresponding distribution function. For the sake of simplicity we will assume that the mean value of the distribution is zero, i.e. that φ'(0) =0. For these calculations we will use the following formula where The dash on the summation sign indicates that the term corresponding to k = 0 is missing.     

On the St. Petersburg Paradox

The classical St. Petersburg Paradox is discussed in terms of doubling strategies. It is claimed that what was originally thought of as a ''paradox'' can hardly be considered as very surprising today, but viewed in terms of doubling strategies, we get some results that look paradoxical, at least to the practically oriented investor.     

Calculating Funding Premiums for Universal Life Insurance

Abstract

This paper provides a brief overview of the most commonly used methods for calculating the level gross premium required to achieve a specific funding target under a typical universal life product design. The first method described utilizes summation techniques from numerical analysis to derive a general accumulation formula. Manipulating this formula leads to an expression for the modal gross funding premium. The second method uses a partial Taylor series expansion to derive a variant of the Newton-Raphson iteration formula, which can be shown to provide second-order convergence to the funding premium. Other iteration methods providing first-order convergence are also described. A more detailed discussion of these methods and their applicability can be found in the author’s paper “Calculating Funding Premiums for Universal Life Insurance, with Examples,” scheduled to be published in the 1999–2000 volume of the TSA Reports.     

Osculatory mechanical quadrature

Abstract

The literature contains many formulas of mechanical quadrature¹, most of which are expressible in the form where the A's are constants, f(a _v) represents the functional value of f(x) at each of the n + 1 points x=a _v (v=0,1,2,..., n), and R is the remainder term. Two general and important types of the above formula are the Newton-Cotes ² formula in which the points a _v are equally spaced from c to d, and the Gaussian ³ quadrature formula in which the a's are chosen so as to obtain the greatest accuracy. The Euler-Maclaurin ⁴ formula of summation and quadrature uses the functional values f(a _v ), and the odd ordered derivatives of f(x) at the end points of the interval of integration. Steffensen ⁵ developed a formula for approximate integration employing not only the functional values but the first derivatives, f'(a _v ).     

Preferences representation in actuarial sciences

Abstract

The paper presents a generalization of Bernoulli's principle and of Von Neumann-Morgenstern's theorem with a view to lending more realism to the representation of preferences in actuarial sciences.     

Truncated inverse binomial sampling

Abstract

1.Introductions Inverse Binomial Sampling.

From an infinite population of a's and b's, in proportions p and q = 1-p respectively, individuals are drawn at random until M a's are found. Thus the sample size, n, is a random variable. Its well-known distribution is      

Deriving the arbitrage pricing theory when the number of factors is unknown

This paper examines the use of proxies (or reference variables) for the true factors in the arbitrage pricing theory (APT). It generalizes other authors' existing work and shows that, when there are more reference variables than the true factors, the APT still holds. The possibility of fewer reference variables than the true factors is also considered, but the APT is not shown to hold, in the same sense, for this case. This work builds on an earlier paper by Ingersoll (Ingersoll J 1984 J. Finance 39 1021-39), and our propositions can be thought of as specializations of his theorems. Similar to Nawalkha (Nawalkha S 1997 J. Financial Economics 46 357-81), our work does not use the mathematics of Hilbert and Banach spaces and, thus, is open to a much wider audience. The practical implication of our results is that model builders should be generous with the number of factors they use, as excessively parsimonious models suffer from inaccuracy.     

Criteria for the Stochastic Ordering of Random Sums,with Actuarial Applications

It is shown that vectors ( S M 1 , … , S Mn ) and ( S' M'1 , …, S' M'n ) of random sums of positive random variables are stochastically ordered by upper orthant dependence, lower orthant dependence, concordance or by the supermodular ordering whenever their corresponding random numbers of terms ( M 1 , … , M n ) and ( M' 1 , … , M' n ) are themselves ordered in this fashion. Actuarial applications of these results are given to different dependence structures for the collective risk model with several classes of business.     

Another pragmatic loss reserving method or Bornhuetter-Ferguson revisited

Abstract

The efficiency of an approximate credibility method for predicting outstanding claims in reinsurance, is analysed. The advantage of the approximate method is, that it does not require exact knowledge of the model's second order moments.     

Time-consistent reinsurance and investment strategies for an AAI under smooth ambiguity utility

This paper investigates time-consistent reinsurance(excess-of-loss, proportional) and investment strategies for an ambiguity averse insurer(abbr. AAI). The AAI is ambiguous towards the insurance and financial markets. In the AAI's attitude, the intensity of the insurance claims' number and the market price of risk of a stock can not be estimated accurately. This formulation of ambiguity is similar to the uncertainty of different equivalent probability measures. The AAI can purchase excess-of-loss or proportional reinsurance to hedge the insurance risk and invest in a financial market with cash and an ambiguous stock. We investigate the optimization goal under smooth ambiguity given in Klibanoff, P., Marinacci, M., & Mukerji, S. [(2005). A smooth model of decision making under ambiguity. Econometrica 73, 1849–1892], which aims to search the optimal strategies under average case. The utility function does not satisfy the Bellman's principle and we employ the extended HJB equation proposed in Björk, T. & Murgoci, A. [(2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance and Stochastics 18(3), 545–592] to solve this problem. In the end of this paper, we derive the equilibrium reinsurance and investment strategies under smooth ambiguity and present the sensitivity analysis to show the AAI's economic behaviors.     

Some results on the compound Markov binomial model

Abstract

Some years ago, in the course of an analysis of upper and lower limits for incomplete moments of statistical distributions I established an elementary summation formula¹ which proved rather useful for the purpose I had in view. Subsequently the formula was generalized by professor Steffensen, who showed² that the formula in question could be looked upon as giving the first term of an expansion in a certain type of series. Professor Steffensen established recurrence formulae for the coefficients of the series and computed the second, third and fourth term and the corresponding remainders¹, but did not arrive at a general, explicite expression for the coefficient of the n-th term and the corresponding remainder. A year later I found these expressions accidentally while I was working on some other problem. I also discovered the real nature of the procedure in question which proved to be a certain kind of least square fitted polynomial approximation. I did not, however, at the time publish the result. Taking the question up again later I found that the whole problem could be considerably generalized. The type of generalization in question is analogous to the generalization from polynomials to arbitrary functions.     

On a generalization of Gautschi's inequality

Abstract

In this note a generalization of Gautschi's inequality satisfied by the gamma function is obtained.     

Profitable technical trading rules as a source of price instability

Abstract

This model incorporates technical trading rules (TTRs) that extract information from the price, allowing the users to benefit from the information. Sustainable profits are possible as long as the price movements reflect changes in the security's intrinsic value. The choice to use the TTR rather than fundamental information is endogenous to the model. Increases in the popularity of the TTR can produce price bubbles and diminish the TTR's ability to extract a reliable signal. Large fluctuations in the TTR's popularity lead to unsustainable periods of positive profits coupled with long-term losses.     

Knowledge Elicitation of Gompertz' Law of Mortality

A new formulation of Gompertz' law of mortality is proposed. Explicit formulas for moment generating function and moments of this formulation are derived. The new formulation is applied to the theory of heterogeneous populations and formulas for stop-loss transformations are derived.     

Public—Private Partnerships in Ireland: Policy and Practice

Ireland's National Development Plan 2000–2006 includes a significant programme of public—private partnerships (PPPs). The Irish Government's policy on PPPs has been shaped to ensure that capital investments under PPP are not included when calculating key fiscal aggregates. This article traces the origins of Ireland's PPP programme and outlines the extent of PPP activity to date. It details how the PPP programme has failed to make an impact in terms of addressing Ireland's infrastructure deficit and examines three particular cases where the PPP model has been applied.     

Performance measurement: The state of the art

Measuring performance is a complex subject, but one which practitioners in the public sector need to get to grips with. What follows is a reader's guide to the available literature and suggestions as to how this should be approached.     

De skandinaviske Aktuarforeningers Virksomhed i 1930

Abstract

The formula II of Mr. Querry's paper may be proved in the following way which also illustrates the real contents of the formula.     

The Effectiveness of PowerPoint presentations in the Accounting Classroom

Abstract

The purpose of this paper is to provide empirical evidence regarding the effectiveness (or not) of using PowerPoint in an accounting classroom. The study focuses on the relationship between students' preferences regarding PowerPoint and their academic performance in the accounting classroom. The study was conducted using the survey method, in which the data was collected from a questionnaire administered to 189 undergraduate students in a medium-size Japanese university. The results of this study show a significant relationship between students' preferences regarding PowerPoint media and their academic performance as shown in their examination scores. Consequently, it was suggested that incorporating multimedia into the accounting classroom does not necessarily provide a simple solution to improving the effectiveness of students' learning outcomes.     

The performance of enhanced-return index funds: evidence from bootstrap analysis

We apply the bootstrap technique proposed by Kosowski et al. [J. Finance, 2006, 61, 2551–2595] in conjunction with Carhart's [J. Finance, 1997, 52, 57–82] unconditional and Ferson and Schadt's [J. Finance, 1996, 51, 425–461] conditional four-factor models of performance to examine whether the performances of enhanced-return index funds over the 1996 to 2007 period are based on luck or superior ‘enhancing’ skills. The advantages of using the bootstrap to rank fund performance are many. It eliminates the need to specify the exact shape of the distribution from which returns are drawn and does not require estimating correlations between portfolio returns. It also eliminates the need to explicitly control for potential ‘data snooping’ biases that arise from an ex-post sort. Our results show evidence of enhanced-return index funds with positive and significant alphas after controlling for luck and sampling variability. The results are robust to both stock-only and derivative-enhanced index funds, although the spread of cross-sectional alphas for derivative-enhanced funds is slightly more pronounced. The study also examines various sub-periods within the sample horizon.     

Facts & figures

The National Audit Office are emerging as a force for change in management reform in the Civil Service. Their pivotal position, poised between Whitehall and Westminster, gives them a responsibility for ensuring that change is informed by the need to retain public support. The Office's validation is important and provides central government with an incentive to get things right.