Communicating a pension product well is as important as optimising the financial value. In a recent study, we showed that up to 80% of the value of a pension lump sum could be lost if customer communication failed. In this paper, we extend the simple customer interaction of the earlier contribution to the more challenging lifetime annuity case. Using a simple mobile phone device, the pension customer can select the life-long optimal investment strategy within minutes. The financial risk trade-off is presented as a trade-off between the pension paid and the number of years the life-long annuity is guaranteed. The pension payment decreases when investment security increases. The necessary underlying mathematical financial hedging theory is included in the study.

This paper follows on from the initial position paper on “The Importance of Biodiversity Risks”, prepared by the Biodiversity and Natural Capital Working party, a volunteer group working under the Sustainability Board. This paper explores the link between zoonotic disease and biodiversity loss and aims to raise awareness and discussion within the actuarial community on why this should be an important consideration in risk management. This paper focuses on how zoonotic diseases emerge, how they are linked to biodiversity loss, the potential impacts in the future and progress within the financial sector. This paper forms part of a collection of papers prepared by volunteers under the Sustainability Board that focus on different elements of biodiversity risk considerations.

The main purpose of this paper is the determination of a basis for the calculation of “prudent” reserves for maturity guarantees in annual and single premium unit-linked life assurance policies, particularly those in which the investment medium is a broad range of U.K. equities. Estimates of the proceeds of equity investments required, and theories of stock market movements are discussed; it is concluded that yearly growth rates, including reinvested net income, may be represented approximately by the lognormal distribution and that yearly growth rates are not random, but negatively correlated. A mathematical model is then developed; the evaluation of the resulting distributions, which is a problem of some interest in its own right in the annual premium case, is discussed in the Appendix. There is a general discussion of certain valuation principles, and we give tables showing the level of reserves suggested by our methods. Various practical questions, such as the treatment of the assets and the effects on reserves of mortality, withdrawals, capital gains tax, policies with different terms and existing business, are also considered.

1.1. The European Economic Community forms the world's largest trading block of countries, accounting for approximately one-third of world trade. Legally it consists of:

• —The European Coal and Steel Community (ECSC); formed 1951.

• —The European Economic Community (EEC); established by the Treaty of Rome in 1957; and

• —The European Atomic Energy Community (EURATOM); established by separate treaty at the same time as the EEC.

The standard table of mortality most generally used in Britain at the present time for life assurance calculations is the A 1924–29 table, which was derived from the experience of assured lives during the period 1924–29. During the quarter of a century that has elapsed since that time there have been substantial changes in mortality rates and the A1924–29 table is to-day out of date. The publication of the A 1924–29 Light table, based upon the experience of certain selected offices whose mortality was lighter than average, has no doubt been helpful in providing a table which goes some way in the direction of the lower mortality rates of to-day. However, this table suffers from the disadvantage that, while the mortality rates at young ages are not low enough for current experience, at some of the older age-groups the rates are too low, with the consequence that the shape of the mortality curve does not accord with present conditions. The greatest proportionate reductions in mortality since 1924–29 have occurred at the younger ages, and there can be little doubt that a new table is needed to provide offices with an efficient and up-to-date instrument for life assurance calculations.

In many problems arising from scientific investigations their theoretical formulation leads to differential equations, the numerical solution of which is frequently an extremely laborious process. To handle such problems various mechanical devices have been developed from time to time and the Differential Analyser is one such device, possessing great flexibility in the range of differential equations which can be dealt with by it. Lord Kelvin conceived of such a machine, and the first practical model was designed by Dr V. Bush of the Massachusetts Institute of Technology. Subsequently another full-scale machine was designed by Prof. D. R. Hartree and is in operation at Manchester University. Various small-scale machines have been constructed in this country and mention may be made of those at London University and Cambridge University, designed by Profs. H. S. W. Massey and J. E. Lennard-Jones respectively.

We will be considering a productive investment project or financial security which yields a sequence of cash flows, positive or negative, over time. Let a1 (dollars) be the cash flow from the project at time t, where t takes the values 0, 1, 2,…, n. Given the known cash flows at from the project, and a known market rate of interest, i per period, at which money may be borrowed or invested, a common procedure is to accept the project if its present value P is greater than zero, where

An internal rate of return of the project is defined to be a solution of the equation

in (− 1, ∞), if, of course, one exists.

The effect of alcohol on the human economy and on the health and prosperity of the community has occupied the close attention of statesmen, physiologists and philanthropists for many generations past, and the question of total abstinence has been probably the most widely discussed social problem for more than half a century. That excess, as compared with moderation in the use of alcohol, is prejudicial in every way is a proposition hardly to be disputed. And many authorities who have given the matter careful consideration contend that even as moderation is superior to excess, so is total abstinence from alcohol superior to moderation. It is the purpose of this paper to examine the truth of this contention in regard to Mortality; with which must be closely associated the questions of health and general well being.

It is well known that a principle of uniform seniority applies to mortality tables which follow Gompertz's or Makeham's law, and, in a modified form, in the case of certain other laws, such as Makeham's second law and the double geometric law. It is natural to inquire what is the most general class of mortality laws to which such a principle applies. In its most general form the uniform seniority principle implies that the value of a joint-life annuity on m lives of different ages is equal to that of a joint-life annuity (possibly computed at a different interest rate) on k lives of equal age, in such a way that the new interest rate i′, the number of substituted lives k, and the difference between the youngest age and the substituted equal age, depend only on the various differences in age between the youngest life and the other original lives, and not on the actual age of the youngest life.

Even within comparatively recent years it was not uncommon to hear the question asked, “What is an Actuary”?; and, before dealing with our subject proper, it would therefore seem necessary, or very desirable at least, to trace to some extent the origin and development, so to speak, of the Actuary, and the rise and progress of the Actuarial Profession.

This paper considers certain applications of economics to actuarial work, concentrating on three issues of public interest. The background to the first two issues is the Government's encouragement of personal pensions particularly through stakeholder pensions. The paper shows how the use of an economics model enables investigations to be made, firstly on the demand for personal pensions by different income groups and secondly on the likely inequality of benefits between generations due to variations in the rates of return. The conclusion on the first issue is that personal pensions are of little benefit to low-earners, i.e. those the Government most wants to benefit, and recommends a way of rectifying this; the main conclusion on the second issue is that households will take action to mitigate much of the inequality. The third issue, that of estimating future rates of return, is one of key financial importance. The paper reviews relevant economic papers in this area and considers how the actuarial profession may improve communication with the economics discipline and with its members.

This paper looks at the problems of assessing, for solvency purposes, the capital requirements of a non-life insurer in the context of the United Kingdom. It considers how these capital requirements might vary according to the different risks to which an insurer is subject and how this Risk-Based Capital (RBC) might be measured in practice, using as a case study the RBC formula recently introduced in the United States of America. The paper also discusses the application of RBC concepts to the problem of internal capital allocation, to assist in measuring an insurer's rate of return to shareholders by business unit, as well as the more obvious regulatory application. The advantages and disadvantages of a formula-based approach to capital requirements for solvency purposes are discussed in comparison with possible alternative approaches to insurance supervision.

This paper considers the market or economic valuation and the hedging of Limited Price Indexed (LPI) liabilities. This involves finding optimal static and dynamic hedging strategies which minimise the riskiness of the investment portfolio relative to the liability.

In this paper we do not aim to find the perfect hedge in a perfect world. Instead, it is assumed that optimisation is restricted to three commonly used asset classes in pension funds: cash; long-term (or irredeemable) fixed-interest bonds; and long-dated index-linked bonds. The economic value of the liability is then defined as the value of the best matching portfolio using a mean/variance type of loss function. Specifically, we adopt the risk minimising approach of Föllmer & Sondermann (1986) and Schweizer & Föllmer (1988). Even with such a simple loss function, establishing the theoretically optimal solution can be difficult. We propose that a practical solution close to the theoretical optimum can be found using two approximations. First, we approximate the ‘true’ stochastic economic model by a vector autoregressive model of order one. Second, we use a sequence of linearisations to approximate non-linear by straightforward quadratic minimisation problems.

The proposed approach is illustrated with various numerical examples, and we compare the results of the approximately optimal hedging strategy with static strategies.

The risk premium on ordinary shares is investigated, by studying the total returns on ordinary shares, and on both long-term and short-term fixed-interest investments over the period 1919 to 1994, and by analysing the various components of that return. The total returns on ordinary shares exceeded those on fixed-interest investments by over 5% p.a. on a geometric mean basis and by over 7% p.a. on an arithmetic mean basis, but it is argued that these figures are misleading, because most of the difference can be accounted for by the fact that price inflation turned out to be about 4.5% p.a. over the period, whereas investors had been expecting zero inflation.

Quotations from contemporary authors are brought forward to demonstrate what contemporary attitudes were. Simulations are used along with the Wilkie stochastic asset model to show what the results would be if investors make various assumptions about the future, but the true model turns out to be different from what they expected. The differences between geometric means of the data and arithmetic means are shown to correspond to differences between using medians or means of the distribution of future returns, and it is suggested that, for discounting purposes, medians are the better measure.

The COVID-19 pandemic creates a challenge for actuaries analysing experience data that include mortality shocks. Without sufficient local flexibility in the time dimension, any analysis based on the most recent data will be biased by the temporarily higher mortality. Also, depending on where the shocks sit in the exposure period, any attempt to identify mortality trends will be distorted. We present a methodology for analysing portfolio mortality data that offer local flexibility in the time dimension. The approach permits the identification of seasonal variation, mortality shocks and occurred-but-not reported deaths (OBNR). The methodology also allows actuaries to measure portfolio-specific mortality improvements. Finally, the method assists actuaries in determining a representative mortality level for long-term applications like reserving and pricing, even in the presence of mortality shocks. Results are given for a mature annuity portfolio in the UK, which suggest that the Bayesian information criterion is better for actuarial model selection in this application than Akaike’s information criterion.

The Contemplation of the Mortality of Mankind, has besides the Moral, its Physical and Political Uses, both which have been some years since most judiciously considered by the curious Sir William Petty, in his Natural and Political Observations on the Bills of Mortality of London, owned by Captain John Graunt. And since in a like Treatise on the Bills of Mortality of Dublin. But the Deduction from those Bills of Mortality seemed even to their Authors to be defective: First, In that the Number of the People was wanting. Secondly, That the Ages of the People dying was not to be had. And Lastly, That both London and Dublin by reason of the great and casual Accession of Strangers who die therein, (as appeared in both, by the great Excess of the Funerals above the Births) rendred them incapable of being Standards for this purpose; which requires, if it were possible, that the People we treat of should not at all be changed, but die where they were born, without any Adventitious Increase from Abroad, or Decay by Migration elsewhere.

Secular variations in longevity and in population aging are of huge interest to actuaries. It is shown here that temporal changes in mortality and natality accompany the recycling of influenza A viruses i.e., the re-exposure of human populations, from time to time, to influenza A viruses antigenically similar to viruses (H1, H2, H3) that circulated in the past. Mortality (and natality) change as birth cohorts (whole population and maternal) with specific types and levels of vulnerability to influenza A re-infections, acquired through early-life effects of infection with one (period-specific) influenza A sub-type, course through subsequent influenza A environments over time. Epidemiologic evidence of association between secular trends in mortality (and natality) and interactions between birth-cohort and period effects of influenza A circulation is presented both for the U.K. and the U.S. New interpretations to several epidemiologic and demographic observations follow from this finding.

In looking over some papers of the late Francis Baily, at the request of his sister, for the purpose of advising on their disposal, I found in one packet a number of letters addressed to him by the late George Barrett, the inventor (in this country) of the method for calculating life contingencies which now goes by his name. I also found the manuscripts of Baily’s three works—on leases, on interest, and on life assurances: these are now in their proper place, the library of the Institute of Actuaries.

From Baily’s methodical habit of arranging his papers, I feel confident that the letters alluded to are all which he received from Barrett: unless, indeed, some passed which he preferred not to preserve. The interest with which actuaries look upon the method published in these letters makes me think it desirable to give some account of them in a work which all actuaries consult.