On the test of the hypothesis that the probability of an event is contained within given limits

Asbtract

The hypernormal (or Lexian) frequency function can be defined by the integral where θ(p) is the frequency (or density) function of p defined in the interval. We have, of course, that and that .     

On the distribution of profits and covering of war-risk

Abstract

Extract

Owing to the low premium rates used by Norwegian Life Insurance Companies, the solution of the question relating to the covering of the war-risk offers, in their case, peculiar difficulties. In their proposal for a new basis of calculations the Norwegian actuaries pointed to the desirability of the Companies trying, to a greater extent than heretofore, to arrive at the full payment, in case of a war, of claims arising from deaths among insured persons participating in the war. Most of the Norwegian Companies introduced the new calculations from Jan. 1st 1925, the proposal of the minority as to loading having been adopted. Only one of the companies, the Idun, granted to its policyholders the right to full indemnity in case of death in war, and this right was given on the basis of a special regulation for the allotment of profits.     

Corporate governance and the probability of default

We employ Merton's probability of default as a continuous ex‐ante measure of the likelihood of firm failure and dynamic panel generalised method of moments to better characterise the relationship between corporate governance and the chance of default. In doing so, we overcome limitations of discrete proxies widely used in previous studies and more completely account for endogeneity issues permeating this area of research. While initial testing designed to facilitate comparison with previous studies suggests a significant relationship between the probability of default and executive pay, board structure and ownership structure, once endogeneity concerns are accounted for, no such relationship remains.     

On a conjecture related to the ruin probability for nonhomogeneous exponentially distributed claims

Recently, some recursive formulas have been obtained for the ruin probability evaluated at or before claim instants for a surplus process under the assumptions that the claim sizes are independent, nonhomogeneous Erlang distributed, and independent of the inter-claim revenues, which are assumed to be independent, identically distributed, following an arbitrary distribution. Based on numerical examples, a conjecture has also been stated relating the order in which the claims arrive to the magnitude of the corresponding ruin probability. In this paper, we prove this conjecture in the particular case when the claims are all exponentially distributed with different parameters.     

On a risk measure inspired from the ruin probability and the expected deficit at ruin

In this paper, we study a risk measure derived from ruin theory defined as the amount of capital needed to cope in expectation with the first occurrence of a ruin event. Specifically, within the compound Poisson model, we investigate some properties of this risk measure with respect to the stochastic ordering of claim severities. Particular situations where combining risks yield diversification benefits are identified. Closed form expressions and upper bounds are also provided for certain claim severities.     

A folding methodology for multivariate extremes: estimation of the spectral probability measure and actuarial applications

In this paper, the folding methodology developed in the context of univariate Extreme Value Theory (EVT) by You et al. is extended to a multivariate framework. Under the usual EVT assumption of regularly varying tails, our multivariate folding allows for the estimation of the spectral probability measure. A new weakly consistent estimator based on the classical empirical estimator is proposed. Its behaviour is illustrated through simulations and an actuarial application relative to reinsurance pricing in the case of an insurance data-set.     

Modeling catastrophic deaths using EVT with a microsimulation approach to reinsurance pricing

Recently, a marked Poisson process (MPP) model for life catastrophe risk was proposed in Ekheden & Hössjer (2014). We provide a justification and further support for the model by considering more general Poisson point processes in the context of extreme value theory (EVT), and basing the choice of model on statistical tests and model comparisons. A case study examining accidental deaths in the Finnish population is provided. We further extend the applicability of the catastrophe risk model by considering small and big accidents separately; the resulting combined MPP model can flexibly capture the whole range of accidental death counts. Using the proposed model, we present a simulation framework for pricing (life) catastrophe reinsurance, based on modeling the underlying policies at individual contract level. The accidents are first simulated at population level, and their effect on a specific insurance company is then determined by explicitly simulating the resulting insured deaths. The proposed microsimulation approach can potentially lead to more accurate results than the traditional methods, and to a better view of risk, as it can make use of all the information available to the re/insurer and can explicitly accommodate even complex re/insurance terms and product features. As an example, we price several excess reinsurance contracts. The proposed simulation model is also suitable for solvency assessment.     

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochasticdifferential equations are consistent and asymptotically efficient,but unfortunately difficult to obtain if a closed-form expressionfor the transitional probability density function of the processis not available. As a result, a large number of competing estimationprocedures have been proposed. This article provides a criticalevaluation of the various estimation techniques. Special attentionis given to the ease of implementation and comparative performanceof the procedures when estimating the parameters of the Cox–Ingersoll–Rossand Ornstein–Uhlenbeck equations respectively.     

Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure

Abstract

In this paper, we consider the optimal proportional reinsurance problem in a risk model with the thinning-dependence structure, and the criterion is to minimize the probability that the value of the surplus process drops below some fixed proportion of its maximum value to date which is known as the probability of drawdown. The thinning dependence assumes that stochastic sources related to claim occurrence are classified into different groups, and that each group may cause a claim in each insurance class with a certain probability. By the technique of stochastic control theory and the corresponding Hamilton–Jacobi–Bellman equation, the optimal reinsurance strategy and the corresponding minimum probability of drawdown are derived not only for the expected value principle but also for the variance premium principle. Finally, some numerical examples are presented to show the impact of model parameters on the optimal results.     

The use of phase-type models for disability insurance calculations

This paper explores the use of phase-type models in actuarial calculations for disability insurance. We demonstrate that the changes in status of disability insureds can be appropriately captured by a phase-type model. Our model represents the aging process as the passage through a number of phases of decreasing vitality. When disabled, individuals additionally pass through several stages that represent duration of disability. Recovery and mortality rates from the earlier stages are greater than those in later stages. Using such a model, explicit and easily calculable expressions are obtained for relevant probabilities and actuarial present values. This facilitates the calculation of premiums and reserves.     

On a nonparametric estimator for ruin probability in the classical risk model

In this paper, we present a nonparametric estimator for ruin probability in the classical risk model with unknown claim size distribution. We construct the estimator by Fourier inversion and kernel density estimation method. Under some conditions imposed on the kernel, bandwidth and claim size density, we present some large sample properties of the estimator. Some simulation studies are also given to show the finite sample performance of the estimator.     

Barrier present value maximization for a diffusion model of insurance surplus

In this paper, we study a barrier present value (BPV) maximization problem for an insurance entity whose surplus process follows an arithmetic Brownian motion. The BPV is defined as the expected discounted value of a payment made at the time when the surplus process reaches a high barrier level. The insurance entity buys proportional reinsurance and invests in a Black–Scholes market to maximize the BPV. We show that the maximal BPV function is a classical solution to the corresponding Hamilton–Jacobi–Bellman equation and is three times continuously differentiable using a novel operator. Its associated optimal reinsurance-investment control policy is determined by verification techniques.     

Automatic balancing mechanisms for notional defined contribution accounts in the presence of uncertainty

The notional defined contribution model combines pay-as-you-go financing and a defined contribution pension formula. This paper aims to demonstrate the extent to which liquidity and solvency indicators are affected by fluctuations in economic and demographic conditions and to explore the introduction of an automatic balancing mechanism (ABM) into the pension scheme. We demonstrate that the introduction of an ABM reduces the volatility of the buffer fund and that, in most cases, the automatic mechanism that re-establishes solvency produces the highest value of the risk-adjusted notional factor.     

A practical solution to the problem of ultimate ruin probability

Abstract

It was the Swiss actuary Chr. Moser who, in lectures at Bern University at the turn of the century, gave the name “self-renewing aggregate” to what Vajda (1947) has called the “unstationary community” of lives, namely where deaths at any epoch are immediately replaced by an equivalent number of births. It was Moser too (1926) who coined the expression “steady state” for the stationary community in which the age distribution at any time follows the life table (King, 1887). With such a distinguished actuarial history, excellently summarized by Saxer (1958, Ch. IV), it behoves every actuary to know at least the definitions and modus operandi of today's so-called renewal (point), or recurrent event, processes.     

Estimating the rank of a beta matrix: a GMM approach1

A full-rank beta matrix is a necessary condition for correctly estimating the risk premia in linear asset pricing models. However, the true values of betas are unobserved in practice and must be estimated. In this paper, we propose a straightforward testing method based on the generalised method of moments to assess whether the beta matrix is of full rank. We show that our method has desirable finite sample properties and performs better than available alternatives. We apply our method to several popular factor models and find that most models have rank deficiency in several datasets.     

On the Efficiency of Conditional Heteroskedasticity Models

This paper discusses how conditional heteroskedasticity models can be estimated efficiently without imposing strong distributional assumptions such as normality. Using the generalized method of moments (GMM) principle, we show that for a class of models with a symmetric conditional distribution, the GMM estimates obtained from the joint estimating equations corresponding to the conditional mean and variance of the model are efficient when the instruments are chosen optimally. A simple ARCH(1) model is used to illustrate the feasibility of the proposed estimation procedure.     

On the test of the correction mechanism for mis-pricing between assets using a statistical yield spread model

This paper investigates the existence of a correction mechanism for mis-pricing between Japanese stock and bond. By this correction mechanism we mean that when deviations occur from the equilibrium levels of the expected return differentials between stock and bond — the risk premium differentials, the market will tend to correct the mis-pricing and bring the expected return differentials back to the equilibrium levels. We assume that the yield spread between the predicted earnings price ratio of stock and the yield to maturity of bond reflects the risk premium between stock and bond, and estimate the equilibrium risk premium differentials and mis-prices between stock and bond by modelling their behaviors with a statistical yield spread model (SYS). Empirical results strongly indicate the existence of the mis-pricing correction mechanism, suggesting the inefficiency of securities markets.