Andrew Cheuk-Yin Ng

Canonical Valuation of Mortality-Linked Securities

A fundamental question in the study of mortality�?linked securities is how to place a value on them. This is still an open question, partly because there is a lack of liquidly traded longevity indexes or securities from which we can infer the market price of risk. This article develops a framework for pricing mortality�?linked securities on the basis of canonical valuation. This framework is largely nonparametric, helping us avoid parameter and model risk, which may be significant in other pricing methods. The framework is then applied to a mortality�?linked security, and the results are compared against those derived from other methods.

Lundberg-Type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin Under the Sparre Andersen Model

Abstract In this paper we consider the Sparre Andersen insurance risk model. Three cases are discussed: the ordinary renewal risk process, stationary renewal risk process, and s-delayed renewal risk process. In the first part of the paper we study the joint distribution of surplus immediately before and at ruin under the renewal insurance risk model. By constructing an exponential martingale, we obtain Lundberg-type upper bounds for the joint distribution. Consequently we obtain bounds for the distribution of the deficit at ruin and ruin probability. In the second part of the paper, we consider the special case of phase-type claims and rederive the closed-form expression for the distribution of the severity of ruin, obtained by Drekic et al. (2003, 2004). Finally, we present some numerical results to illustrate the tightness of the bounds obtained in this paper.

On the Upcrossing and Downcrossing Probabilities of a Dual Risk Model With Phase-Type Gains

In this paper, we consider the dual of the classical CramA©r-Lundberg model when gains follow a phase-type distribution. By using the property of phase-type distribution, two pairs of upcrossing and downcrossing barrier probabilities are derived. Explicit formulas for the expected total discounted dividends until ruin and the Laplace transform of the time of ruin under a variety of dividend strategies can then be obtained without the use of Laplace transforms.

Pricing and Hedging Variable Annuity Guarantees with Multiasset Stochastic Investment Models

Variable annuities are often sold with guarantees to protect investors from downside investment risk. The majority of variable annuity guarantees are written on more than one asset, but in practice, single-asset (univariate) stochastic investment models are mostly used for pricing and hedging these guarantees. This practical shortcut may lead to problems such as basis risk. In this article, we contribute a multivariate framework for pricing and hedging variable annuity guarantees. We explain how to transform multivariate stochastic investment models into their risk-neutral counterparts, which can then be used for pricing purposes. We also demonstrate how dynamic hedging can be implemented in a multivariate framework and how the potential hedging error can be quantified by stochastic simulations.

Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model

In this paper, we consider a Markov-modulated risk model (also called Markovian regime switching insurance risk model). Follow Asmussen (2000, 2003), by using the theory of Markov additive process, an exponential martingale is constructed and Lundberg-type upper bounds for the joint distribution of surplus immediately before and at ruin are obtained. As a natural corollary, bounds for the distribution of the deficit at ruin are obtained. We also present some numerical results to illustrate the tightness of the bound obtained in this paper.

Stochastic life table forecasting: A time-simultaneous fan chart application

Given a fitted stochastic mortality model, we can express the uncertainty associated with future death rates in terms of confidence or prediction intervals. Recently, a group of researchers have proposed using fan charts to display prediction intervals for future mortality rates. Existing mortality fan charts are based on isolated pointwise prediction intervals. By pointwise we mean that the interval reflects uncertainty in a quantity at a single point of time, but it does not account for any dynamic property of the time-series. In this paper, we overcome this limitation by introducing the concept of time-simultaneous fan charts. In more detail, instead of pointwise intervals, a time-simultaneous fan chart is derived from a prediction band with a prescribed probability of covering the whole time trajectory. We present two numerical methods for producing time-simultaneous fan charts. These methods can be applied to common stochastic mortality models, including the generalized Cairns–Blake–Dowd model. Finally, the proposed method is illustrated with mortality data from the populations of Australia and New Zealand.

A note on the strong approximation for long memory processes and its application

Let {X k ; k≥1} be a long memory process defined by where {ϵ i ;−∞<i<∞} is a doubly infinite sequence of independent and identically distributed random variables and a i ∼ i −α l(i) for some 1/2<α<1 is a sequence of real numbers. Under some mild conditions, a general strong approximation theorem for partial sums of {X k ; k≥1} is derived, where the variances of the innovations may be infinite. As applications, we establish a general law of the iterated logarithm for the long memory processes and investigate the asymptotic properties of the heavy-tailed long memory model and the adjusted range of partial sums for a kind of long memory processes.

Dettweiler, Egbery, 2004,Risk Processes, EAGLE-Lecture, Leipzig: edition am Gutenbergplatz Leipzig: 207 pages, Eur 19.50, ISBN 3-937219-08-0/pbk

Most books on risk theory start with the simplest classical model and then proceed to more complicated risk models with different interclaim distributions. Claims at different time points are assumed to be independent in most of the models. Thus, the risk process is generated by an independently marked point process. This book takes a different point of view and treats the risk process as a general marked point process with possibly dependent marks. The term ‘‘risk process’’ used in the book only refers to the time and amount of claims. Thus it is somewhat different from the more popular understanding that a risk process is a surplus process. The book contains five chapters and an appendix. They are:

“The Time Value of Ruin in a Sparre Andersen Model,’ Hans U. Gerber and Elias S. W. Shiu, April 2005

ANDREW C. Y. NG* I wish to congratulate Professors Gerber and Shiu for this interesting paper, which studies the ordinary Sparre Andersen model when the interclaim times are independent and distributed as in (3.1). Here I want to illustrate how the results can be extended to the stationary renewal risk model. First we consider a special Sparre Andersen model with the following structure on the interclaim times: