The new technical basis of the Swedish life insurance companies

Abstract

During the past year most of the Swedish life insurance companies have agreed upon instituting common technical basis for the calculation of premiums, policy reserves, surrender values and prospective bonus (returns of premiums). Chiefly the same basis also has been adopted by two companies outside the agreement, and with one exception all Swedish life insurance companies now use the same premiums.     

A proposal for a new basis of life insurance calculations in Norway

Abstract

The basis used by Norwegian life insurance companies for the calculation of premiums and reserves is, as regards insurances, The Institute of Actuaries Life Tables H ^M (20 British Life Offices tables) with interest at 4 per cent. This basis has been used by Norwegian companies for more than thirty-five years and has proved entirely safe.     

Insurance economics

Abstract

Background

Insurance accounting is generally speaking based upon the idea that a comparison shall be made between “premiums earned” and “claims incurred”. Even if there are exceptions in different countries and in different classes of business the method where premiums earned and claims incurred are compared is so widely used that we will take this method as our starting point for a discussion of the shortcomings, if any, of insurance accounting.     

Sickness experience of Framtiden

Abstract

Introduction.

Livförsäkringsbolaget Framtiden, ömsesidigt (the Life Insurance Co. “The Future”, Mutual), to the policyholders of which I have the honour of devoting my actuarial work, is a relatively new enterprise. It was founded in 1911. Its main work is industrial life insurance with monthly premiums. All industrial policies include the waiver of premiums on disablement by sickness or accident for at least four weeks. Disablement lasting more than four weeks involves the waiver of premiums from the beginning of the disablement. For industrial policies issued in late years, the payment of premiums is limited to 65 years of age. Besides ordinary endowment (or whole life) policies, many childrens' insurances are written, including some or other insurance of the bread-winner, at all events the waiver of premiums till the 20th birthday of the child, if the supporter should die before that time, and for the periods during this time when the supporter is disabled by sickness or accident for at least four weeks. It has been tried to propagate more protective forms of children's policies, but it has partly failed, owing to lack of interest from agents and public. Nevertheless, it has been possible to exclude pure childrens' tariffs, and so in almost every policy issued, there is a moment of sickness insurance. In late years, this holds true also for the ordinary branch. The mass of experience collected is thus by no means unimportant, despite the smallness of the Company, as compared with foreign Companies. The experience is, moreover, collected during a relatively short period of time, giving a rather homogeneous material.     

Remarks on risk theory

§ 1. The Scheme U(n).

Following remarks are the result of some deliberations which I have made in order to find a simple starting point for rough estimates of the risk reserves necessary under given conditions. They do not appear with any pretensions. The idea, upon which my thoughts were based, is far from original: I have hoped to be able to simplify the theory by simplifying in a high degree the hypotheses and by confining myself to small numbers and arithmetical methods.     

Studies in insurance mathematics

Abstract

In the new Rules of Calculation of the Life Insurance Co. Framtiden, the old idea of the continuous mode of payment has been realized. In practice, this method only signifies, that the premium is to be restored for the time elapsed after the moment of death. Theoretically, it makes unnecessary the computation of premiums by the aid of (exact or approximate) yearly, half-yearly, quarterly and monthly annuities, continuous annuities being solely needed. It may perhaps be of interest to give a more detailed account of the method employed.     

Risk theory in a Markovian environment

Abstract

We consider risk processes _{t t?0} with the property that the rate β of the Poisson arrival process and the distribution of B of the claim sizes are not fixed in time but depend on the state of an underlying Markov jump process {Z_t } _t?0 such that β=β _i and B=B_i when Z_t=i . A variety of methods, including approximations, simulation and numerical methods, for assessing the values of the ruin probabilities are studied and in particular we look at the Cramér-Lundberg approximation and diffusion approximations with correction terms. The mathematical framework is Markov-modulated random walks in discrete and continuous time, and in particular Wiener-Hopf factorisation problems and conjugate distributions (Esscher transforms) are involved.     

On finite-time ruin probabilities for classical risk models

Abstract

Generally, the return of premiums without interest or with simple interest is provided for. It might, however, be worth while to notice that less complicated formulre are needed for the return of premiums with compound interest.     

Equity and Credibility

We build on previous work concerned with measuring equity and consider the problem of using observed claim data or other information to calculate premiums which maximize equity. When these optimal premiums are used, we show that gathering more information or refining the risk classification always increases equity. We study the case for which the premium is constrained to be an affine function of the claim data and obtain results analogous to classical credibility theory, including the inhomogeneous and homogeneous cases of the Bu¨hlmann-Straub model. We derive formulas for the credibility weights in certain cases.     

Bounds for Actuarial Present Values Under the Fractional Independence Assumption

We consider the fractional independence (FI) survival model, studied by Willmot (1997), for which the curtate future lifetime and the fractional part of it satisfy the statistical independence assumption, called the fractional independence assumption.

The ordering of risks of the FI survival model is analyzed, and its consequences for the evaluation of actuarial present values in life insurance is discussed. Our main fractional reduction (FR) theorem states that two FI future lifetime random variables with identical distributed curtate future lifetime are stochastically ordered (stop-loss ordered) if, and only if, their fractional parts are stochastically ordered (stop-loss ordered).

The well-known properties of these stochastic orders allow to find lower and upper bounds for different types of actuarial present values, for example when the random payoff functions of the considered continuous life insurances are convex (concave), or decreasing (increasing), or convex not decreasing (concave not increasing) in the future lifetime as argument. These bounds are obtained under the assumption that some information concerning the moments of the fractional part is given. A distinction is made according to whether the fractional remaining lifetime has a fixed mean or a fixed mean and variance. In the former case, simple unique optimal bounds are obtained in case of a convex (concave) present value function.

The obtained results are illustrated at the most important life insurance quantities in a continuous random environment, which include bounds for net single premiums, net level annual premiums and prospective net reserves.     

Upper bounds on stop-loss premiums under constraints on claim size distribution

Abstract

The problem of maximal stop-loss premium under prescribed constraints on claim size distribution is taken up again. The methods of linear programming are used to show that the recent results of others are intuitively obvious. These results are then extended by the linear programming technique to cases of more general constraints, e.g. prescribed claim size variance, or prescribed minimum frequency of excess claims. In particular it is shown that, typically, the upper bound on stop-loss premiums is generated by a claim size distribution which has all its mass concentrated at very few points. In contrast with the results obtained by others recently, it is seen that the claim size distribution which produces the maximal stop-loss premium is not generally independent of the excess. Some numerical examples are given showing that the methods used here can sometimes improve considerably the recent results of others. The case of a compound Poisson distribution is treated briefly.     

Myopic loss aversion and margin of safety: the risk of value investing

Abstract

Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this paper we develop a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow, and analyse it using simulation, dimensional analysis, and theoretical tools based on mean field approximations. The model makes testable predictions for basic properties of markets, such as price volatility, the depth of stored supply and demand versus price, the bid–ask spread, the price impact function, and the time and probability of filling orders. These predictions are based on properties of order flow and the limit order book, such as share volume of market and limit orders, cancellations, typical order size, and tick size. Because these quantities can all be measured directly there are no free parameters. We show that the order size, which can be cast as a non-dimensional granularity parameter, is in most cases a more significant determinant of market behaviour than tick size. We also provide an explanation for the observed highly concave nature of the price impact function. On a broader level, this work suggests how stochastic models based on zero intelligence agents may be useful to probe the structure of market institutions. Like the model of perfect rationality, a stochastic zero intelligence model can be used to make strong predictions based on a compact set of assumptions, even if these assumptions are not fully believable.     

General approximation schemes for option prices in stochastic volatility models

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.     

Stochastic Orderings of Convex-Type for Discrete Bivariate Risks

New classes of order relations for discrete bivariate random vectors are introduced that essentially compare the expectations of real functions of convex-type of the random vectors. For the actuarial context, attention is focused on the so-called increasing convex orderings between discrete bivariate risks. First, various characterizations and properties of these orderings are derived. Then, they are used for comparing two similar portfolios with dependent risks and for constructing bounds on several multilife insurance premiums.     

A Note on Gaussian Estimation of the CKLS and CIR Models with Feedback Effects for Japan

In this note we extend the Gaussian estimation of two factor CKLS and CIR models recently considered in Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34) to include feedback effects in the conditional mean as was originally formulated in general continuous time models by Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) with constant volatility. We use the exact discrete model of Bergstrom, A. R. (1966, Non-recursive models as discrete approximations to systems of stochastic differential equations, Econometrica 34, 173–182) to estimate the parameters which was first used by Brennan, M. J. and Schwartz, E. S. (1979, A continuous time approach to the pricing of bonds, J. Bank. Financ. 3, 133–155) to estimate their two factor interest model but incorporating the assumption of Nowman, K. B. (1997, Gaussian estimation of single-factor continuous time models of the term structure of interest rates, J. Financ. 52, 1695–1706; 2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34). An application to monthly Japanese Euro currency rates indicates some evidence of feedback from the 1-year rate to the 1-month rate in both the CKLS and CIR models. We also find a low level-volatility effect supporting Nowman, K. B. (2001, Gaussian estimation and forecasting of multi-factor term structure models with an application to Japan and the United Kingdom, Asia Pacif. Financ. Markets 8, 23–34).     

Magisterkonferens i Forsikringsmatematik ved Københavns Universitet 1922–23. Den skriftlige Prøve.

Abstract

A. Purpose of this note

Any specific application of the theory in Section 3 of the paper would demand that the statewise reserves V_j be precisely defined. There is some latitude at this point, however, and it turns out that Theorem 3 as stated may require that an appropriate definition be used. Paragraph B of the present note adds rigour on the issue. Paragraph C offers some guidance as to how to construct and compute the reserves in nontrivial cases. Some technical lemmas are placed in the final Paragraph D.     

Bayesian Premium Rating with Latent Structure

We propose a fully Bayesian approach to non-life risk premium rating, based on hierarchical models with latent variables for both claim frequency and claim size. Inference is based on the joint posterior distribution and is performed by Markov Chain Monte Carlo. Rather than plug-in point estimates of all unknown parameters, we take into account all sources of uncertainty simultaneously when the model is used to predict claims and estimate risk premiums. Several models are fitted to both a simulated dataset and a small portfolio regarding theft from cars. We show that interaction among latent variables can improve predictions significantly. We also investigate when interaction is not necessary. We compare our results with those obtained under a standard generalized linear model and show through numerical simulation that geographically located and spatially interacting latent variables can successfully compensate for missing covariates. However, when applied to the real portfolio data, the proposed models are not better than standard models due to the lack of spatial structure in the data.     

Some results on moments and cumulants

Abstract

In the present paper we discuss various results related to moments and cumulants of probability distributions and approximations to probability distributions. As the approximations are not necessarily probability distributions themselves, we shall apply the concept of moments and cumulants to more general functions. Recursions are deduced for moments and cumulants of functions in the form R_k [a, b] as defined by Dhaene & Sundt (1996). We deduce a simple relation between the De Pril transform and the cumulants of a function. This relation is applied to some classes of approximations to probability distributions, in particular the approximations of Hipp and De Pril.     

Hedge Accounting and its Consequences on Portfolio Earnings – A Simulation Study

Abstract

In this paper, I analyze the consequences of cash flow hedge accounting on portfolio earnings of firms focusing on main changes between IFRS 9 and IAS 39. For this purpose, I develop a simulation study which illustrates the quantitative effects on the accounting entries according to the currently applicable hedge accounting methods. It is especially addressed what accounting differences arise and how these distinctions may affect a firm’s earnings. Furthermore, I examine to which firms early switching becomes especially desirable or burdensome. This information is particularly useful to managers and investors. The paper shows that portfolio earnings are affected differently. In the model, IAS 39 may lead to higher or lower earnings for increasing deviations between foreign and domestic interest rates. Additionally, sensitivity to volatility changes varies among the methods. Moreover, a partly ineffective hedging relationship does not necessarily decrease earnings compared to its fully effective counterpart.     

An optimal stopping problem in risk theory

Abstract

In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.