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.     

V. E. Gamborg

Abstract

A glance at the numerous papers dealing with the influence of the rate of interest on the value of premiums will show that most authors aim at computing annuity values for a new rate of interest without first re-calculating the commutation columns, It is only in exceptional cases that they derive premiums or policy-values directly, i.e. without first finding annuity values.² Apart from the fact, that both premiums and policyvalues are implicitly given by a set of annuity values, the reason for the usual procedure lies in the type of calculations which is contemplated, since they cannot conveniently be applied to such ratios as premiums and policy-values. In the following lines we show how the method developed by A. J. Lotka ³ for the calculation of the rate of increase of a stable population is capable of generalisation and of successful application to our problem; thus the detour via the calculation of annuity values can be avoided.     

Studies in risk theory with numerical illustrations concerning distribution functions and stop loss premiums. Part I

Abstract

The aim of this paper is to analyse two functions that are of general interest in the collective risk theory, namely F, the distribution function of the total amount of claims, and II, the Stop Loss premium. Section 2 presents certain basic formulae. Sections 17-18 present five claim distributions. Corresponding to these claim distributions, the functions F and II were calculated under various assumptions as to the distribution of the number of claims. These calculations were performed on an electronic computer and the numerical method used for this purpose is presented in sections 9, 19 and 20 under the name of the C-method which method has the advantage of furnishing upper and lower limits of the quantities under estimation. The means of these limits, in the following regarded as the “exact” results, are given in Tables 4-20. Sections 11-16 present certain approximation methods. The N-method of section 11 is an Edgeworth expansion, while the G-method given in section 12 is an approximation by a Pearson type III curve. The methods presented in sections 13-16, and denoted AI-A4, are all applications and modifications of the Esscher method. These approximation methods have been applied for the calculation of F and II in the cases mentioned above in which “exact” results were obtained. The results are given in Tables 4-20. The object of this investigation was to obtain information as to the precision of the approximation methods in question, and to compare their relative merits. These results arc discussed in sections 21-24.     

Nils Ekholm

Abstract

The problem of χ² tests of a linear hypothesis H₀ for ‘matched samples’ in attribute data has been discussed earlier by the author (Bennett, 1967, 1968). This note presents corresponding results for the hypothesis that the multinomial probabilities p satisfy (c ?1) functional restrictions: F ₁(p) = 0, ... , F _C?1(p) = 0. An explicit relationship between the usual ‘goodness-of-fit’ χ² and the modified minimum χ² (=χ*²) of Jeffreys (1938) and Neyman (1949) is demonstrated for this situation. An example of the test for the 2 × 2 × 2 contingency table is given and compared with the solution of Bartlett (1935).     

Generic pricing of FX,inflation and stock options under stochastic interest rates and stochastic volatility

We consider the pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility, for which we use a generic multi-currency framework. We allow for a general correlation structure between the drivers of the volatility, the inflation index, the domestic (nominal) and the foreign (real) rates. Having the flexibility to correlate the underlying FX/inflation/stock index with both stochastic volatility and stochastic interest rates yields a realistic model that is of practical importance for the pricing and hedging of options with a long-term exposure. We derive explicit valuation formulas for various securities, such as vanilla call/put options, forward starting options, inflation-indexed swaps and inflation caps/floors. These vanilla derivatives can be valued in closed form under Schöbel and Zhu [Eur. Finance Rev., 1999, 4, 23–46] stochastic volatility, whereas we devise an (Monte Carlo) approximation in the form of a very effective control variate for the general Heston [Rev. Financial Stud., 1993, 6, 327–343] model. Finally, we investigate the quality of this approximation numerically and consider a calibration example to FX and inflation market data.     

Consistency and asymptotic normality of maximum likelyhood estimators

Abstract

Let the random variable X denote the time taken in completion of a process. For a fixed a, if the observed value of X is less than a, the X is observable, but if X is greater than a, the process is tampered with and is accelerated or decelerated at time a by some unknown factor α, and Y=a+α(X-a) is observed. If the experimenter has only partial control over the experiment, it may be difficult to get several observations on Y corresponding to the same a value. Thus we have a set of independent but not identically distributed observations. The large sample behavior of m.l.e. of the unknown parameters based on tampered random variables Y ^b1 , ..., Y ^bn is studied. If X follows an exponential distribution with mean (1/--), ... the consistency and asymptotic normality of the m.l.e. of α and -- is established under mild conditions on a ^b1, a ^b2, ... The conditions needed for establishing the consistency of m.l.e. of lX are given when X follows a uniform distribution U(O, --) or when X has any known distributional form     

Least squares,and a generalisation of the »Student»-Fisher theorem

Abstract

1. Following the methods employed by Steffensen ¹ a simple analytical proof of a generalisation of the »Student»-Fisher theorem is obtained. Geometrical proof of this generalisation has been given by Sterne.²     

Cohort and value-based multi-country longevity risk management

ABSTRACT

Multi-country risk management of longevity risk provides new opportunities to hedge mortality and interest rate risks in guaranteed lifetime income streams. This requires consideration of both interest rate and mortality risks in multiple countries. For this purpose, we develop value-based longevity indexes for multiple cohorts in two different countries that take into account the major sources of risks impacting life insurance portfolios, mortality and interest rates. To construct the indexes we propose a cohort-based affine model for multi-country mortality and use an arbitrage-free multi-country Nelson–Siegel model for the dynamics of interest rates. Index-based longevity hedging strategies have the advantages of efficiency, liquidity and lower cost but introduce basis risk. Graphical risk metrics are a way to effectively capture the relationship between an insurer's portfolio and hedging strategies. We illustrate the effectiveness of using a value-based index for longevity risk management between two countries using graphical basis risk metrics. To show the impact of both interest rate and mortality risk we use Australia and the UK as domestic and foreign countries, and, to show the impact of mortality only, we use the male populations of the Netherlands and France with common interest rates and basis risk arising only from differences in mortality risks.     

A Recursive Scheme for Perpetuities with Random Positive Interest Rates. II: The Impenetrable Wall

In some former contributions, the authors investigated actuarial quantities with stochastic interest rates. In a first model, the randomness is modelled by means of an ordinary Wiener process, and as a consequence negative interest rates are possible. A second model provides a tool to avoid these negative interest rates, which can be necessary in particular situations. This paper wants to present an alternative solution to the problem of negative interest rates. This new model will be implemented to the case of an annuity certain and of a perpetuity.     

Bank Exposure to Interest Rate Risks During Financial Liberalization: Evidence from South Korea

This study documents the changing impact of long and short term interest rate risks on the equity prices of banks in South Korea during the process of financial liberalization. Consistent with the presence of regulatory constraints, Korean bank equity returns are found to be sensitive to both anticipated and unanticipated changes in interest rates in the first period (1976-81) when banks were largely under government control. However, during our last period (1989-99) of liberalization, Korean bank equity returns were found to have a positive association only with unanticipated short-term interest rates. Consistent with the ability to manage other interest rate risks successfully, in this last liberalization period, Korean bank equity returns had no association with long-term or with anticipated short-term interest rates. In view of the continued interest in banking and financial market liberalization among many Asian, African, and formerly socialist countries including China, these results should be of much banking and policy interest. JEL Classifications: G21, G28, E44, L89     

Stochastic interest rate modelling using a single or multiple curves: an empirical performance analysis of the Lévy forward price model

In current financial markets negative interest rates have become rather persistent, while in theory it is often common practice to discard such rates as incredible and irrelevant. However, from a risk management perspective, it is crucially important to financial institutions to properly account for this phenomenon in their Asset Liability Management (ALM) studies. In this paper, we develop a coherent framework on how to best incorporate negative interest rates in these studies through a single curve stochastic term structure model and compare it to its multiple curve analogue. It turns out that, from the wide range of available single curve models, especially the Lévy Forward Price model (LFPM) of Eberlein and Özkan [The Lévy LIBOR model. Financ. Stoch., 2005, 9, 327–348] seems appropriate for ALM purposes. This paper describes an optimisation routine for calibrating this LFPM under the risk-neutral measure in both the single and multiple curve framework to the market prices of interest rate caplets with different strike rates, maturities and tenors. In addition, an empirical performance analysis is made of the single and multiple curve LFPM, where we include four deterministic volatility specifications and provide an explicit parametrisation of a piecewise homogeneity restriction with both deterministic and random breakpoints. This comparative analysis indicates that both the single and multiple curve LFPM is best adopted with the Linear-Exponential Volatility (LEV) specification and that deterministic breakpoints should be included, rather than random breakpoints.     

Assessing the Time-Varying Interest Rate Sensitivity of Real Estate Securities

Abstract

This paper examines the sensitivity of real estate securities to changes in both market and central bank interest rates. It is commonly viewed that the traded real estate market is one of the industry sectors most susceptible to interest rate movements. This is due to traditional high levels of borrowing, the impact of rate changes on property yields and indirectly upon occupational demand and thus rental income. The results which are the first to examine the UK sector, highlight the impact of interest rates on UK property companies, in relation to both returns and volatility. The paper also illustrates that this sensitivity is not confined to periods of high and volatile interest rates as the sample period under examination is characterized by historically low and stable rates.     

The representation of American options prices under stochastic volatility and jump-diffusion dynamics

This paper considers the problem of pricing American options when the dynamics of the underlying are driven by both stochastic volatility following a square-root process as used by Heston [Rev. Financial Stud., 1993, 6, 327–343], and by a Poisson jump process as introduced by Merton [J. Financial Econ., 1976, 3, 125–144]. Probability arguments are invoked to find a representation of the solution in terms of expectations over the joint distribution of the underlying process. A combination of Fourier transform in the log stock price and Laplace transform in the volatility is then applied to find the transition probability density function of the underlying process. It turns out that the price is given by an integral dependent upon the early exercise surface, for which a corresponding integral equation is obtained. The solution generalizes in an intuitive way the structure of the solution to the corresponding European option pricing problem obtained by Scott [Math. Finance, 1997, 7(4), 413–426], but here in the case of a call option and constant interest rates.     

A Generalization of the Formulas for Options on the Maximum or the Minimum of Several Assets

Abstract

This paper generalizes the option on the maximum or the minimum of two assets (several assets) within a stochastic interest rate framework. A Gaussian model is used to describe the interest rates. Closed-form solutions for the market values are presented. The use of the options is illustrated with numerical examples.     

Stationarity aspects of the sparre andersen risk process and the corresponding ruin probabilitles

Abstract

The Sparre Andersen risk model assumes that the interclaim times (also the time between the origin and the first claim epoch is considered as an interclaim time) and the amounts of claim are independent random variables such that the interclaim times have the common distribution function K(t), t|>/ 0, K(O)= 0 and the amounts of claim have the common distribution function P(y), - ∞ < y < ∞. Although the Sparre Andersen risk process is not a process with strictly stationary increments in continuous time it is asymptotically so if K(t) is not a lattice distribution. That is an immediate consequence of known properties of renewal processes. Another also immediate consequence of such properties is the fact that if we assume that the time between the origin and the first claim epoch has not K(t) but as its distribution function (k^b1 denotes the mean of K(t)) then the so modified Sparre Andersen process has stationary increments (this works even if K(t) is a lattice distribution).

In the present paper some consequences of the above-mentioned stationarity properties are given for the corresponding ruin probabilities in the case when the gross risk premium is positive.     

On minimum variance stratification for estimating the mean of a Weibull population

Abstract

The problem of dividing the frequency function of the Weibull distribution into L(L = 1, ... ,6) strata for the purpose of estimating the population mean under optimum allocation from a stratified random sample is considered. The optimum points of stratification (y ¹,..., y^L-1 ) determining the minimum variance of the estimator are obtained. The variance of the sampling units in each stratum and the variance of the estimate are also given.     

On the Gerber-Shiu Discounted Penalty Function for the Ordinary Renewal Risk Model with Constant Interest

Abstract

In this paper we study the Gerber-Shiu discounted penalty function for the ordinary renewal risk model modified by the constant interest on the surplus. Explicit answers are expressed by an infinite series, and a relational formula for some important joint density functions is derived. Applications of the results to the compound Poisson model are given. Finally, a lower bound and an upper bound for the ultimate ruin probability are derived.     

Negative real interest rates

Standard textbook general equilibrium term structure models such as that developed by Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407], do not accommodate negative real interest rates. Given this, the Cox, Ingersoll, and Ross [1985b. “A Theory of the Term Structure of Interest Rates.” Econometrica 53 (2): 385–407] ‘technological uncertainty variable’ is formulated in terms of the Pearson Type IV probability density. The Pearson Type IV encompasses mean-reverting sample paths, time-varying volatility and also allows for negative real interest rates. The Fokker–Planck (i.e. the Chapman–Kolmogorov) equation is then used to determine the conditional moments of the instantaneous real rate of interest. These enable one to determine the mean and variance of the accumulated (i.e. integrated) real rate of interest on a bank (or loan) account when interest accumulates at the instantaneous real rate of interest defined by the Pearson Type IV probability density. A pricing formula for pure discount bonds is also developed. Our empirical analysis of short-dated Treasury bills shows that real interest rates in the UK and the USA are strongly compatible with a general equilibrium term structure model based on the Pearson Type IV probability density.     

Pricing defaultable bonds: a middle-way approach between structural and reduced-form models

In this paper we present a valuation model that combines features of both the structural and reduced-form approaches for modelling default risk. We maintain the cause and effect or ‘structural’ definition of default and assume that default is triggered when a state variable reaches a default boundary. However, in our model, the state variable is not interpreted as the assets of the firm, but as a latent variable signalling the credit quality of the firm. Default in our model can also occur according to a doubly stochastic hazard rate. The hazard rate is a linear function of the state variable and the interest rate. We use the Cox et al. (A theory of the term structure of interest rates. Econometrica, 1985, 53(2), 385–407) term structure model to preclude the possibility of negative probabilities of default. We also horse race the proposed valuation model against structural and reduced-form default risky bond pricing models and find that term structures of credit spreads generated using the middle-way approach are more in line with empirical observations.     

The DB Underpin Hybrid Pension Plan

Abstract

In this paper the defined benefit underpin guarantee is valued as a financial option, within the traditional funding paradigms of actuarial science. Assuming fixed interest rates, and assuming that salaries can be treated as a tradable asset, we value the guarantee using fair value principles. Contribution rates are developed for the Entry Age Normal, Projected Unit Credit, and Traditional Unit Credit funding methods. In addition, for the accruals methods, we demonstrate the implied hedging strategy. The traditional unit credit offers the best method of these three, as it is consistent with the principles of financial economics, and the resulting contributions more naturally follow the cost of the emerging benefit, without creating expensive barriers to new hires. The method generates significant contribution volatility, and we demonstrate how this can be reduced with suitable benefit design and ongoing risk management.