On the solution of a functional equation concerning mortality formulae

Abstract

1. If (x) and (y) are lives whose remaining lifetimes are stochasticallyindependent, and if the mortality of each of the lives is given by a Makeham expression, then as a well known fact (see e.g. P. F. Hooker & L. H. Longley-Cook, Life and Other Contingencies, Cambridge 1957, vol. II, pp. 137&138) the evaluation of joint-life endowments and joint-life annuities on the lives (x) and (y) may be performed by substituting a single life (u) for (x) and (y) and altering the force of interest, provided that and with the same value of the parameter c( > 1).     

Large sample validity of the binomial distribution for lives with unequal mortality rates

Abstract

Let us consider a group of n lives which are observed during some time or age interval. Suppose that the following conditions are satisfied: 1. The probability of death within the interval considered has the same value q for each person of the group.

2. These lives represent statistically independent observations (with respect to mortality).

     

Actuarial validity of the binomial distribution for large numbers of lives with small mortality probabilities

Abstract

In a previous paper (see reference [1]), the binomial distribution was shown to be valid under rather general conditions for the case of a large number of statistically independent lives with possibly unequal mortality probabilities. This note extends these results to some situations where the lives are not necessarily statistically independent. An analysis is presented which indicates that these situations include most actuarial applications involving a large number of lives.     

On a characterization of geometric distribution

Abstract

Recently Lyle Cook (1978) has obtained a characterization of the exponential distribution. We obtain an analogue of this result for geometric distribution.     

On the distribution and estimation of trading costs

This paper investigates the uncertainty about the trading costs associated with a given portfolio strategy. I derive accurate approximations of the ex ante probability distributions of proportional trading costs and portfolio turnover under the conventional assumption of normal asset returns. Based on these approximations, I express the expected trading costs as a function of asset and portfolio characteristics. All else equal, the expected trading costs increase with: i) the deviations of the expected asset returns from the expected portfolio return, ii) the assets' volatility and iii) the portfolio volatility. At the same time, they decrease with the covariance between the assets and the portfolio. Furthermore, I propose novel estimators of the expected turnover and trading costs and show that they offer small bias and low variance, even when the sample size is small. Finally, I incorporate my results into a portfolio selection framework to produce portfolios with low levels of risk and trading costs. Several experiments with real and simulated data confirm the practical value of the results.     

A note on the efficiency of the binomial option pricing model

We discuss the efficiency of the binomial option pricing model for single and multivariate American style options. We demonstrate how the efficiency of lattice techniques such as the binomial model can be analysed in terms of their computational cost. For the case of a single underlying asset the most efficient implementation is the extrapolated jump-back method: that is, to value a series of options with nested discrete sets of early exercise opportunities by jumping across the lattice between the early exercise times and then extrapolating from these values to the limit of a continuous exercise opportunity set. For the multivariate case, the most efficient method depends on the computational cost of the early exercise test. However, for typical problems, the most efficient method is the standard step-back method: that is, performing the early exercise test at each time step.     

On reaching for yield and the coexistence of bubbles and negative bubbles

We develop a model of financial intermediation wherein bank managers “reach for yield” – by overinvesting in risky assets and underinvesting in safer assets – provided they do not face much cost from liquidity shortfalls. The managers follow a pecking order in which their first preference is to invest in risky assets; their second preference is to hoard liquid assets; and their last preference is to invest in safer assets. This behavior is conducive to the formation of bubbles and “negative” bubbles in the market for risky and safer assets, respectively. Monetary loosening, by reducing the cost of liquidity shortfalls, induces further reach for yield and amplifies the bubbles.     

On the optimum spacing of sample quantiles from a normal distribution. Part I

Abstract

Assume that a large number of observations are made on a normal random variable with the density function where σ σ 0, When the sample is very large the ordinary estimates of µ and a involve considerable computational work. In order to simplify the estimation of µ and/or σ it is sometimes convenient to select a small number of sample quantiles and to use estimates which are linear functions of these sample quantiles, Such a procedure is particularly convenient when the observations occur naturally in order of magnitude, which happens in life testing, for instance, Let      

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.     

Pricing and managing risks of European-style options in a Markovian regime-switching binomial model

We discuss the pricing and risk management problems of standard European-style options in a Markovian regime-switching binomial model. Due to the presence of an additional source of uncertainty described by a Markov chain, the market is incomplete, so the no-arbitrage condition is not sufficient to fix a unique pricing kernel, hence, a unique option price. Using the minimal entropy martingale measure, we determine a pricing kernel. We examine numerically the performance of a simple hedging strategy by investigating the terminal distribution of hedging errors and the associated risk measures such as Value at Risk and Expected Shortfall. The impacts of the frequency of re-balancing the hedging portfolio and the transition probabilities of the modulating Markov chain on the quality of hedging are also discussed.     

On moments of order statistics from the pareto distribution

Abstract

The evaluation of multiple integrals which occur in order statistics distribution theory is involved due to the fact that the integration is to be carried on over an ordered range of variables of integration. This difficulty is sometimes completely obviated by transforming the ordered variates to the unordered ones. Several such transformations are available in the Theory of Multiple Integrals. In previous papers [2, 3] the author used one such transformation, and gave alternative simplified proofs of several known results in the distribution theory of order statistics from the exponential and the power function distributions. In this paper we use such a known transformation to derive moments (and distributions if necessary) of order statistics from the Pareto distribution. Malik [4] has derived moments of order statistics from this distribution without the transformation of the ordered variates to the unordered ones. The process of direct integration used by Malik becomes complicated for dealing with the moments of more than two ordered variates. Further, the method which we use here is unformly applicable to derive the moments or the distributions of one or more ordered variables, and gives the distributions and moments without any complicated steps in integration. The transformation used by us considerably simplifies the manipulations necessary for the derivation of moments or the Mellin transforms, and thus we hope that our paper would at least be of Pedagogical interest.     

On the asymptotic distribution of weighted least squares estimators

Abstract

In the ELB (Empirical Linear Bayes)-approach to credibility, the unknown structural parameters are substituted by a set of parameter estimates. The weighted least squares estimators are known to be asymptotically normally distributed when the design variables are independent and identically distributed random variables. It is demonstrated that, with probability one, the conditional asymptotic distribution, given the design, is the same as the unconditional distribution. Estimation of the asymptotic covariance matrix will also be considered.     

On evaluation of the Delaporte distribution and related distributions

Abstract

In the present paper we develop recursive algorithms for evaluation of the Delaporte distribution, the compound Delaporte distribution, and convolutions of compound Delaporte distributions. Some asymptotic results are given. We discuss how the approach can sometimes be generalized to other classes of compound mixed Poisson distributions when the mixing distribution is a shifted infinitely divisible distribution.     

On the (in)feasibility of covered interest parity as a solution to the forward bias puzzle

This paper examines Pippenger's (2011) proposed solution to the forward bias puzzle, which is based on the covered interest parity (CIP) condition. It is argued that the CIP-based approach does not solve this well known and long-standing puzzle in international finance in a meaningful way. Moreover, it is shown that empirical results from such an approach follow mechanically from the identity-like nature of the theory of covered interest parity, which, aside from small deviations due to transaction costs, is assumed to hold in all periods (as if it were an identity). We show that rather than leading to new insights, the simple reconfiguration of CIP to solve for the time t + 1 spot exchange rate leads to tautological expressions that, when estimated, might appear to successfully explain the forward bias, but in actuality are trivial. Results from simple simulation exercises further illustrate the inconclusiveness of the proposed solution method.     

Achieving smooth asymptotics for the prices of European options in binomial trees

A new binomial approximation to the Black–Scholes model is introduced. It is shown that, for digital options and vanilla European call and put options, a complete asymptotic expansion of the error in powers of n ^?1 exists. This is the first binomial tree for which an asymptotic expansion has been shown to exist.     

On the independence of system life distribution and cause of failure

Abstract

Under the competing risks model, we obtain conditions for and consequences of the independence of the system life length and the cause of failure. When the survival distributions are continuous, we consider the situations where the risks are independent as well as they are dependent. In the dependent case, the discussion is limited to two risks with some special bivariate survival distributions. The discussion of discrete model where we assume the survival distributions to be discrete, is limited to two independent risks. This results in two characterizations of geometric distribution. Finally some generalizations of our results to k out of m systems are considered.     

On the distribution of government bond returns: evidence from the EMU

This paper assesses the statistical distribution of daily EMU bond returns for the period 1999–2012. The normality assumption is tested and clearly rejected for all European countries and maturities. Although skewness plays a minor role in this departure from normality, it is mainly due to the excess kurtosis of bond returns. Therefore, we test the Student’s t, skewed Student’s t, and stable distribution that exhibit this feature. The financial crisis leads to a structural break in the time series. We account for this and retest the alternative distributions. A value-at-risk application underlines the importance of our findings for investors. In sum, excess kurtosis in bond returns is essential for risk management, and the stable distribution captures this feature best.     

On the distribution of discounted loss reserves using generalized linear models

Renshaw and Verrall [] specified the generalized linear model (GLM) underlying the chain-ladder technique and suggested some other GLMs which might be useful in claims reserving. The purpose of this paper is to construct bounds for the discounted loss reserve within the framework of GLMs. Exact calculation of the distribution of the total reserve is not feasible, and hence the determination of lower and upper bounds with a simpler structure is a possible way out. The paper ends with numerical examples illustrating the usefulness of the presented approximations.     

On Tail Value-at-Risk for sums of non-independent random variables with a generalized Pareto distribution

Recently in actuarial literature several authors have derived lower and upper bounds in the sense of convex order for sums of random variables with given marginal distributions and unknown dependency structure. In this paper, we derive convex bounds for sums of non-independent and identically distributed random variables when marginal distributions are mixture models. In particular, we examine some well-known risk measures and we find approximations for Tail Value-at-Risk of the sums considered when marginal distributions are generalized Pareto distributions. By numerical examples we illustrate the goodness of the presented approximations.